UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2015 …€¦ · THREE (3) QUESTIONS. ALL QUESTIONS...
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TITLE OF PAPER COURSE CODE TIME ALLOWED REQUIRMENTS INSTRUCTIONS UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2015 NON-PARAMETRIC ANALYSIS ST409 2 (TWO) HOURS STATISTICAL TABLES AND CALCULATOR ANSWER QUESTION ONE AND ANY THREE (3) QUESTIONS. ALL QUESTIONS CARRY MARKS AS INDICATED WITHIN THE PARENTHESIS. THIS PAPER IS NOT TO BE OPENED UNTIL PERMISSION HAS BEEN GRANTED BY THE INVIGILATOR
Transcript of UNIVERSITY OF SWAZILAND MAIN EXAMINATION PAPER 2015 …€¦ · THREE (3) QUESTIONS. ALL QUESTIONS...
TITLE OF PAPER
COURSE CODE
TIME ALLOWED
REQUIRMENTS
INSTRUCTIONS
UNIVERSITY OF SWAZILAND
MAIN EXAMINATION PAPER 2015
NON-PARAMETRIC ANALYSIS
ST409
2 (TWO) HOURS
STATISTICAL TABLES AND CALCULATOR
ANSWER QUESTION ONE AND ANY THREE (3) QUESTIONS ALL QUESTIONS CARRY MARKS AS INDICATED WITHIN THE PARENTHESIS
THIS PAPER IS NOT TO BE OPENED UNTIL PERMISSION HAS BEEN GRANTED BY THE INVIGILATOR
Page I ANSWER QUESTION ONE amp ANY THREE QUESTIONS
For all questions clearly state the name ofthe test type ofthe test (upper lower or two-sided) the null amp alternate hypotheses the test statistics the decision rule the level ofsignificance
the decision amp the conclusions
QUESTION ONE [ 20 + 20 marks]
Fifteen volunteers for an experiment are divided randomly into three groups to see if telescopic sight improves the ability to hit the target under twilight conditions Group A is given rifles with telescopic sights (auto focus) group B is given same kind of rifles with telescopic sights but with manual focus while group C has the same kind of rifle with open sights After a reasonable learning period they are given a shooting test at twilight These are their scores (100 is perfect)
Group A 96 93 88 85 89 GroupB 92 97 87 91 96 Groupe 89 93 80 77 84
a Use an appropriate test to test whether any difference exists between the scores of auto and manual focus Use 5 level ofsignificance
b Use another appropriate test to determine whether evidence exists to conclude that the assistance in sights tend to differ in hitting targets Use (l =010 k
QUESTION TWO [ 10 + 10 marks]
A According to the director of a tourist bureau there is a median of 10 hours of sunshine per day during the summer months For a random sample of 20 days during the past three summers the number ofhours of sunshine has been recorded as shown below
8 9 8 10 9 7 7 9 7 7 9 8 11 9 10 7 8 11 8 12
Test the directors claim at 5 level of significance using an appropriate tests based on the binomial distribution Also calculate the P-value
B The following table shows the information on cotton crop insurance for the years 1976 to 2000
Of Crops ofCrops Year insured Year insured 1976 19479 1989 15375 1977 26667 1990 21312 1978 63969 1991 26526 1979 57715 1992 24865 1980 38086 1993 21152 1981 38434 1994 23458 1982 24196 1995 25774 1983 19319 1996 32646 1984 29975 1997 31786 1985 25451 1998 24821 1986 20410 1999 19593 1987 19940 2000 14960 1988 15628
Do these data indicate a downward trend in the number of crops insured Use the Cox-Stuart test for trend with (l = 005
Page 2
OUESTION THREE [20 marks]
Two judges rated the participants in a dance contest as fol1ows
Contestant
Judge A B C D E F G H
1 5 2 6 3 4 8 7
2 3 1 7 4 5 2 6 8
Are the two rankings independent Use either Spearmans p test or Kendalls T test with (l =005
OUESTION FOUR [20 marks]
Two computer software packages are being considered for use in the inventory control department ofa small manufacturing firm The firm has selected 12 different computing tasks that are typical of the kinds ofjobs such a package would have to perform and then recorded tle number of seconds each package required to complete each task given in the following table Do the data present sufficient evidence to indicate that both softwares performed the jobs at the same time Analyse the data by using the Wilcoxon Signed Rank Test with (l = 010
Computing Time Required for Software Task Packages X and Y
A B C D E F G H I J K L
240 167 216 237 375 314 149 373 179 155 290 199
231 204 177 207 421 361 218 403 260 155 354 255
OUESTION FIVE [20 marks]
A random sample of five sixth-grade boys in one section of town were given a literacy test with the following results 82 74 87 86 75 Another random sample ofeight sixth-grade boys from a different section of town were given the same literacy test with these scores reSUlting 88 77 9188949383 94 Using Smimov test find whether there is a difference in literacy in the two populations of sixthshygrade boys
i
TABLE AI Nonnal Distributionshy
SeJecred wIuu uooI -37190 z- - -32905 ZuIs -19600 Zu -16449
r- 37190 z-shy 32905 ZuIs = 19600 Zushy 16449 tIIIII 1001 1002 10D3 1104 1005 11116 l1li7 1111 1119
IDD -3D9Ol -2I7Bl -27478 -265l1 -25758 -25121 -24573 -2laquo189 -2J656
111 -23263 -U904 -22571 -22262 -21973 -21701 -21+14 -212111 -2OH9 -20749
112 -20S37 -lD315 -201-41 -19954 -1977-4 -19600 -190431 -19268 -19110 -18957
113 -18B08 -18663 -18522 -18384 -1B250 -18119 -17991 -17866 -177 -1762-4
104 -17507 -17392 -17279 -17169 -17060 -16954 -16849 -16747 -16646 -16546
1115 -16449 -16l52 -16258 -16164 -16072 -15982 -15893 -15805 -15718 -15632
1116 -15548 -15464 -15382 -15301 -152211 -151-41 -15063 -1-4 -11909 -104833
117 -14758 -14684 -1461 -1-4538 -1-4466 -14395 -14325 -um -14187 -14118
1M -14051 -1J9IM -13917 -1J8Sl -1J787 -1J722 -13658 -US95 -13532 -13469
119 -Il408 -1J346 -1328S -13225 -1J165 -1J106 -13047 -12_ -12930 -12873
111 -12816 -12759 -Il702 -12646 -12591 -12536 -120481 -12426 -12372 -12319
111 -12265 -12212 -12160 -12107 -12055 -12004 -11952 -11901 -11850 -11800
112 -11751 -11700 -11650 -11601 -11552 -11503 -1165 -11407 -11359 -IIU I
113 -11264 -11217 -11170 -11123 -11077 -1I0U -1D98S -1D939 -10B93 -1111MB
114 -11803 -10758 -lml-4 -IoK9 -1D625 -Iosal -10537 -1G4H -1M50 -111407
115
I117
-10364 -09945 -09542
-1G322 -09904 -D9SQ2
-1079 -09863 -09463
-1G237 -091122 -oMlltt
-101904 -o97B2 -09385
-10152 -09741 -09346
-10110 -09701 -09307
-IJI06 -09661 -09269
-IDOl7 -09amp21 -09230
-09986 -o95BI -09192
118 -09154 -09116 -09078 -09040 -09002 -DlI6S -DB927 -D8B90 -08BS3 -D8B16
119 -08779 -D8742 -D8705 -D8669 -08633 -DBS96 -DI56O -D85lltt -oampt8B -08452
UI -D8416 -DB3BI -0836 -DB310 -0827-4 -08239 -DB21l-4 -DBI69 -18134 -oB099
121 -0B064 -0JI030 -07995 -07961 -D7926 -07892 -07858 -D7B2-f -07790 -01756
122 -07722 -07688 -17655 -07621 -07588 -D7554 -07521 -07_ -07454 -07421
123 -07388 --07356 -07323 -D7290 -D7257 -07225 -07192 -07160 -07128 -07095
124 -07063 -07031 -06999 -06967 -06935 -06903 -D6871 -o6lMO -D6B08 -06776
TABLE AI (Continued)
IIDO 0101 1002 1003 11 OIOS 01116 0007 1001 1009015 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -D6S26 -D6495 -06464 126 -06433 -06403 -06372 -06341 -06311 -062BO -06lS0 -D6219 -06189 -06158 027 -06128 -06098 -06068 -D6038 -06008 -05978 -G5Ha -o591B -05888 -05858 028 -O5B28 -05799 -05769 -D574O -05710 -05681 -05651 -05622 -05592 -05563 029 -0553-4 -05505 -05476 -DS446 -05-417 -05388 -05359 -05330 -05302 -05273 030 -05244 -05215 -05187 -05158 -05129 -05101 -05072 -05044 -05015 -04987 031 -04959 -04930 -04902 -04874 -0-486 -004817 -04789 -04761 -04733 -0-4705 032 -04677 -004649 -04621 -0-4593 -04565 -04538 -04510 -044112 -04454 -Q4427 033 -04399 -04372 -04344 -04316 -04289 -01261 -04234 -0207 -0-4179 -04152 034 -04125 -0-4097 -0-4070 -04043 -0-4016 -03989 -03961 -03934 -03907 -03880 ClE -03853 -03826 -0379 -O3m -03745 -0371 -03691 -Q3665 -03638 -0361 I 111 nlI
-03585 -0331~
-035515 -onn
-0lS3[ -03266
-03505 -03239
-0gt478 -0Z213
-03451 -O pound6
-0H2S -O~IpoundgtO
-03398 -0031gt4
-03372 -01107
-033( -030SI
tlJe -03055 -O302~ -03002 -O2~ipound -021750 -O2ni -02pound -02871 -O2E -02217 ~O9 -O27~3 -027t -027lt 1 -01715 -0261lt -026pound1 -02637 -02GI [ -025pound5 -O25~
IlAO -0253 -02508 -02482 -02~b -02lt10 -O240 -O237a -121S2~ -a22T -O2~O[
1il41 -02275 -OlliO -0222( -02191gt -02173 -021lt7 -02121 -02096 -02070 -02045 112 -02019 -01913 -01968 -01942 -Ol~r7 -0le91 -01866 -0114O -01815 -01789 O4l -01764 -01738 -01713 -01687 -01662 -0637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -01231 -01206 -01181 -01156 -01130 -01105 -0 lOBO -01055 -01030 046 -01004 -00979 -00954 -00929 -0G904 -00878 -00853 -00828 -00B03 -00778 047 -00753 -00728 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0G476 -0G451 -00426 -00401 -00376 -00351 -00326 -00301 -00276 049 -00251 -00226 -00201 -00175 -00150 -00125 -00100 -00075 -00050 -OoOlS 050 00000 OoolS 00050 00075 00100 OOllS 00150 00175 002111 00226 051 00251 00276 00301 00326 00351 00376 00401 0G426 0G451 00476 ts2 00502 00527 00552 00577 00602 0D627 0G6S2 00677 00702 00728 053 00753 00778 00803 0G82B 00853 0G87B 0090-4 00929 0095-4 00979 054 01004 01030 01055 OIOBO 01105 01130 01156 01181 01206 01231
c
I ampII
Tabla AI (Continued)
1- IDDI 1DD2 UDJ D8D4 IDDS 1DD6 U07 ICIOI UDt
155 01257 01282 01307 01332 oll58 oll83 ola 01304 01 59 DIo484 156 01510 01535 01560 01586 01611 0 I 6rT 01662 01687 01711 Dl718 D57 01764 01789 01815 01840 01866 01891 01917 01942 01968 0 I 99l 151 DlOI9 02fM5 02070 112096 02111 021-47 02173 02198 0122 f1225O 059 02215 02l01 02327 O2lSJ 02l71 02-404 02 10 02-456 02C 02508 161 O25ll 0l559 02585 02611 026l7 0266l Dl689 02715 1U71 02767 161 0219) 02819 02845 02B71 D289II 02924 02950 02976 03002 D1029 162 03055 03081 03107 031304 03160 03186 0311l 0J2l9 0l266 D32n 161 03319 033045 O3m 03398 030415 03451 U47I 03505 o3SlI Dl558 D6ltt 0358S 03611 03638 03665 0369l 03719 03745 urn cum 03826 us 03853 038110 03907 039304 03961 03989 0-4016 0-4043 0-4070 0-4097 G66 025 0151 0-4179 0 bull Q07 0-41304 0161 0219 0-4316 O l U371 D67 Ul99 044l7 0 5-4 0+182 04510 0 538 0565 04593 011 0-4649 I6D 04677 0705 0m3 0-4761 0789 017 0-48-45 0-487 0 901 0910 169 0-4959 0987 osol5 oso+f O507l 05101 051l9 05158 05187 05115 17D 05l+i D5l73 D5302 05310 05359 05_ oHI7 05+t6 05-476 ossos D71 055304 05563 05592 D56ll 05651 05681 D5710 D57-40 05769 05799 In 058l8 05858 05888 05918 059-48 05978 D6OO8 06038 D6068 D6098 Dn 06128 06158 06189 06l19 II6l5O 06280 06311 063041 DU71 D6403 17-4 06-433 06-464 060495 G65l6 06557 06588 D6QD 06amp51 0668l 06713 US 067 5 06776 06808 D684O 06871 06903 06935 06967 06999 07031 176 07063 07095 07128 07160 07192 07l25 onsJ Ono 07311 07356 D77 07388 0721 0745-4 OJ D7511 Oj55-4 07588 07611 07655 07688 D7I 07711 07756 07790 07114 07858 071192 079l6 07961 07995 08010 D 011064 08099 0813 08169 08204 08l3J 08l74 08310 0835 08l81 DM 08416 08452 0 oas24 08560 oas96 08633 08669 08705 D87-42 111 08779 08816 08853 08890 D8917 08965 09002 090-40 09071 09116 Da2 0915-4 09192 G92lO 09169 09307 09346 G9385 09-424 09463 09502
Table AI (ContInued)
DG IoDI DoD2 DDOl D LIDS DDD6 DDD7 1001 11D9
Da3 095-42 09581 09621 09661 G9701 09741 09782 09822 09863 09904 DoN 09945 09986 10027 1D069 10110 10152 1019gt4 10237 10279 10322 DaS 10364 10407 10450 1049gt4 10537 IOS81 1D6l5 10669 1071 10758 186 10803 10848 10893 10939 10985 11031 11077 11123 11170 11217 0B1 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 lUi 11750 lI80() Ll85() 11 9() I 1I~52 12004 12055 12107 2160 12212 alll~middot pound2265 IUIS [nn 12426 L24S1 2S36 f25~[ 12~6 [ 2702 L175~
QIQ 12Bt6 12Em i2nD L29BB t3M7 1310( 13165 13l25 32E5 131-~C
SS[ IHO~ ~E~ [m poundi57$ 365C ln2 117CT r 3eS2 1717 f7l rt
~~2 [4Q[1 I~I( lmiddotmmiddotT flt2ES IQI f bull tJpound 1ltlt66 Ilt~ [A~II i 468lt U [i7S [4823 [lt9(1~ [Age~ S(l(~ 15[lt[ 1522( [5301 (36 ~4H Qbull( 155ltamp [51gt32 [5718 15005 15e5~ 19gt8 [(072 16[(( [6258 [63S2 Ui L~49 16S46 LeuroG46 167lt7 1(849 [695-1 17060 l7169 17279 73n 4) 17507 176l4 7741l 17866 17991 1amp1 9 18250 1pound384 [8522 1l63 cn 18808 1B157 19110 19268 19431 100 1977-4 19954 20141 20335 098 20537 20749 20969 21201 21444 21701 21973 22262 22571 22904 U9 23263 23656 2-4089 24573 25121 25758 26521 2778 28782 30902
SouRCE Generated by IL L man Used with permission
bull The emries In this table are quamiIes ~ of the standard normal random YIlIiabIe Z eIeted iO P(Z s tJ - and P(Z gt tJ - I - p Note that the value of co two decimal pIacas determines which raw co use the third decimal pIKe of Pdetermines which column to use to find Zp
ampII
=
510 APPeNDIX APPENDIX 511
TABLEAl Chl-Squd Dlatrlbutlonmiddot TABLEA3 Binomial DIstribution
0750 otoO 0950 0975 0990 ot95 9 n I 005 010 015 01pound1 elgt 010 015 00 05
k =I 1 1 5 6 7 8 9
10 II 11 13 15 16 17 18 19 10 11 11 11 1 15 16 17 11 19 10 ltto SO 60 70 80
100 Z
UU 1773 lt4108 5385 6626 78011 9037
10n 1139 1155 1370 1lt485 1598 1711 IUS 1937 10lt49 1160 lln 1381 193 1604 27 281 291 303 3153 3262 3371 380 62 5633 amp698 7758 8813 9865
lot1 0675
1706 605 6251 7779 9136
10M 1101 1116 1lt468 1599 1718 1855 1981 1106 nll 235lt4 1lt477 1599 1710 18lt41 1961 lUI 3101 3UO 318 3556 367lt4 3791 39ot lt40l6 5181 6311 7lt40 8553 9658
1076 1185
281
31111 5991 7815 9lt188
1107 1159 107 1551 1691 1831 1968 1103 1136 2368 1500 lUO 1759 2887 301 31 1 3167 1191 3517 36lt42 3765 3889 lt4011 1301 4156 377 5576 6750 7908 to53
IOU 1131 1143
16lt15
501 7378 9]f8
111 1l8l 1 5 1601 1753 1902 lM8 1191 23]f 17 l6ll 17lt49 1885 1019 3153 3185 3417 354 3618 3808 1917 lt4065 lt4192 119 lt4sn 698 593 71lt41 8330 9502
1066 1181 1296
1960
6635 9110
113 1328 1501 1681 184 1009 1167 lUI 1lt473 16ll 1769 1 3058 1100 33lt11 11 3619 3757 lB93 lt4029 16lt1
~8 31 lt4564 -96 ltI8l8 959 SU 6369 1615 8838
10M 1111 111 1358
1326
717 1060 1114 16 1675 1855 lUI 113n 1519 2676 1830 111 31]1 ]110 1lt417 nn 1716 3158 -4000 I 110 1 556 -93 4l9
4so 5134 5367 6677 19lt4 195
1042 IIU 1183 10101
lS76
1083 1381 1617 18017 1051
n 1lt431 1611 1788 195 lr16 1191 3Ul 1611 3770 315 lt4079 131 381 4532 lt4410 17 973 5118 5161 05 55 561 5130 5970 73 IU6 9961
1111 128 1371 14
3090
1
3
5
6
7
0
0 I 1
0
1 1
0 I 1 1 lt4
0 I 1 1 4 5
0 I 1 1 5 6
0 I 1 3 4
09500 10000
09025 09975 10000
0857lt4 09928 09999 10000
081lt45 09860 09995 10000 10000
07738 09714 09988 10000 10000 10000
07351 096n 09918 09999
10000 10000 10000
06983 09556 09962 09998 10000
09000 10000
08100 09900 10000
0n9O uno 09990 10000
06561 09477 09963 09999 10000
05905 09185 0991lt4 09995 10000 10000
0531 Q8857 09842 09981 09999 10000 10000
0lt4183 08501 097lt43 09973 09998
08500 10000
07225 09775 10000
06141 09392 09966 10000
05220 08905 09880 09995 10000
04431 08352 09134 09918 09999 10000
03171 01165 09527 09941 09996 10000 10000
03206 01166 09262 09879 09988
OBOOO 10000
06400 000 10000
OSlili
00960 09910 10000
010) OBI92 O912(J 099114 10000
0317 07373 09411 099D 09997 10000
02611 0655[ 09011 09830 09984 O99) 10000
01097 05767 03510 09667 09953
llSOO 10000
t)5~25
o~37S
1000u (H21~
oll43a 09841 10000
021 161 omlJ 0I92 091 10000
on73 00)28 OflS 09044 09990 10000
o171l0 05339 011306 014 09954 099WI 10000
01335 D4middot-H9 07564 O92N 09071
07000 10000
04900 09100 10000
03430 07840 09730 10000
02401 06511 09163 09919 10000
01681 052B2 08369 09692 09916 10000
01116 0A202 01443 09295 09B91 09993 10000
00824 03294 06411 08140 09712
06500 10000
04225 08775 10000
02146 01182 09511 10000
01185 05630 08735 09850 10000
01160 U284 01648 09460 09941 10000
00154 01191 06411 08826 09117 09982 10000
00490 02338 05323 08002 09444
Q6000 10000
03600 08400 10000
02160 06480 09360 10000
01296 0lt4152 08208 09144 10000
00778 01310 06826 09130 09898 10000
00461 02333 05443 08208 09590 09959 10000
00280 01586 04199 01102 09037
05500 10000
03025 07975 10000
01664 05148 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00277 01636 04415 07447
09308 09911 10000
00152 01024 01164 06083 08471
for gt 100 lUI lba ~ w - (i)(z + Vlk shy 1)1 or tIM monI ~ w -
Ie (I shy ~+ z ft) wh Z II tIM wlua from tIM _danIIud lIOIIIIII dllaiblldaft IIhown In lba bouom
5 6 7
10000 10000 10000
10000 10000 10000
09999 10000 10000
099 10000 10000
09997 09999 10000
09962 09998 10000
09910 09994 10000
09812 09984 10000
09643 09963 10000
of lba IIIbIa ScuIcE Abridpd from Tabla S VoL I of Puraon and Hanlay (191) wllb parmlNlon from tIM ~ Tru_ bullTha IllIriu In lbl tabla ara quantIIeIl of II chi-lIIl11n1d random wrlalH W with du of rndom bullbull IecuMlIO P(W gtIi W) - Pand P(W gt - I - P
512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
II 050 155 060 065 070 075 010 085 G90 095I
0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
I 0999 09917 09661 O91J-l 08343 07297 06089 04826 Ol 10000 09991 09944 0980--1 (9511 09012 08283 07l3-4 062 0 00625 00-410 00256 00150 00081 00039 00016 00005 00001 00000
I 03125 025 01791 01265 00837 00501 00272 00120 00037 00005 5 10000 09999 09994 099 09900 0977 0ff60f 09006 08l22 06875 06090 052 0370 0383 02617 01808 01095 00523 001-40 6 10000 10000 10000 099) 09991 09957 09888 09750 095023 09375 09085 0870-4 01215 07599 06836 059Oltt 0-4780 03-439 01855 7 10000 10000 10000 10000 U999l 09996 09986 09962 09909 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 8 10000 10000 10000 10000 10000 10000 09999 09997 09991
J 0 00312 00185 00102 00053 0002 00010 00003 00001 00000 00000 9 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 01875 01312 00870 005-40 00308 00156 00067 ooon 00005 00000 10 0 05987 03-487 01969 (JI07-J 0OS63 00282 00ll5 00060 00025 2 05000 0-4069 0317 02352 01631 01035 00579 00266 00086 00012 I G9139 07361 05443 03750 u2-l4() 0193 00860 00-464 002l3 3 08125 0738 06630 05716 0718 03672 01627 016-48 00815 0Gl26 1 09885 09298 08202 06778 05156 03828 02616 01673 00996 4 09688 09-497 onn 088-40 08319 01627 06723 05563 0-4095 02262 J 09990 09872 09500 Qrl79 I (J71S) 06496 05138 03823 02660 J 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 09999 0998-4 09901 072 09119 0B497 07515 06lll 050-4-4
6 0 00156 00083 000-41 00018 00007 00G02 00001 00000 00000 00000 5 10000 09999 09986 099) u9il03 09527 09051 083l8 07l8-4 1 OlOff 00692 bull 0G410 oom 00109 00G46 00016 0000-4 00001 00000 6 10000 10000 09999 09991 09965 098 097-4Q 090452 089BO l 0338 02553 01791 0117 00705 00376 0G170 00059 00013 00001 7 10000 10000 10000 0999) 099 09984 09952 09877 09726 3 06562 05585 M557 03529 02557 016904 00989 00-473 00158 00022 8 10000 10000 10000 10000 I()OOO 09999 09995 09983 09955 08906 0836-4 07667 06809 05789 0-4661 U446 onl5 03 00328 9 10000 10000 10000 10000 10000 10000 10000 09999 09997 J 09ampM 09723 09533 0916 0882 uno 07379 06229 04686 026-49 10 10000 10000 10000 00011 10000 10000 10000 10000 10000
6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
6 urn 0980f8 09720 G9510 09176 08665 07903 0679-f 05217 03017 8 10000 10000 10000 10000 09999 0999-4 09980 01 098527 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 10000 10000 10000 09998 0999l 09978
10 10000 10000 10000 10000 10000 10000 10000 10000 09998 II 10000 10000 10000 10000 10000 10000 10000 10000 10000
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IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
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0000 I OQoII
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3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
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16 17
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1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
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TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
Page I ANSWER QUESTION ONE amp ANY THREE QUESTIONS
For all questions clearly state the name ofthe test type ofthe test (upper lower or two-sided) the null amp alternate hypotheses the test statistics the decision rule the level ofsignificance
the decision amp the conclusions
QUESTION ONE [ 20 + 20 marks]
Fifteen volunteers for an experiment are divided randomly into three groups to see if telescopic sight improves the ability to hit the target under twilight conditions Group A is given rifles with telescopic sights (auto focus) group B is given same kind of rifles with telescopic sights but with manual focus while group C has the same kind of rifle with open sights After a reasonable learning period they are given a shooting test at twilight These are their scores (100 is perfect)
Group A 96 93 88 85 89 GroupB 92 97 87 91 96 Groupe 89 93 80 77 84
a Use an appropriate test to test whether any difference exists between the scores of auto and manual focus Use 5 level ofsignificance
b Use another appropriate test to determine whether evidence exists to conclude that the assistance in sights tend to differ in hitting targets Use (l =010 k
QUESTION TWO [ 10 + 10 marks]
A According to the director of a tourist bureau there is a median of 10 hours of sunshine per day during the summer months For a random sample of 20 days during the past three summers the number ofhours of sunshine has been recorded as shown below
8 9 8 10 9 7 7 9 7 7 9 8 11 9 10 7 8 11 8 12
Test the directors claim at 5 level of significance using an appropriate tests based on the binomial distribution Also calculate the P-value
B The following table shows the information on cotton crop insurance for the years 1976 to 2000
Of Crops ofCrops Year insured Year insured 1976 19479 1989 15375 1977 26667 1990 21312 1978 63969 1991 26526 1979 57715 1992 24865 1980 38086 1993 21152 1981 38434 1994 23458 1982 24196 1995 25774 1983 19319 1996 32646 1984 29975 1997 31786 1985 25451 1998 24821 1986 20410 1999 19593 1987 19940 2000 14960 1988 15628
Do these data indicate a downward trend in the number of crops insured Use the Cox-Stuart test for trend with (l = 005
Page 2
OUESTION THREE [20 marks]
Two judges rated the participants in a dance contest as fol1ows
Contestant
Judge A B C D E F G H
1 5 2 6 3 4 8 7
2 3 1 7 4 5 2 6 8
Are the two rankings independent Use either Spearmans p test or Kendalls T test with (l =005
OUESTION FOUR [20 marks]
Two computer software packages are being considered for use in the inventory control department ofa small manufacturing firm The firm has selected 12 different computing tasks that are typical of the kinds ofjobs such a package would have to perform and then recorded tle number of seconds each package required to complete each task given in the following table Do the data present sufficient evidence to indicate that both softwares performed the jobs at the same time Analyse the data by using the Wilcoxon Signed Rank Test with (l = 010
Computing Time Required for Software Task Packages X and Y
A B C D E F G H I J K L
240 167 216 237 375 314 149 373 179 155 290 199
231 204 177 207 421 361 218 403 260 155 354 255
OUESTION FIVE [20 marks]
A random sample of five sixth-grade boys in one section of town were given a literacy test with the following results 82 74 87 86 75 Another random sample ofeight sixth-grade boys from a different section of town were given the same literacy test with these scores reSUlting 88 77 9188949383 94 Using Smimov test find whether there is a difference in literacy in the two populations of sixthshygrade boys
i
TABLE AI Nonnal Distributionshy
SeJecred wIuu uooI -37190 z- - -32905 ZuIs -19600 Zu -16449
r- 37190 z-shy 32905 ZuIs = 19600 Zushy 16449 tIIIII 1001 1002 10D3 1104 1005 11116 l1li7 1111 1119
IDD -3D9Ol -2I7Bl -27478 -265l1 -25758 -25121 -24573 -2laquo189 -2J656
111 -23263 -U904 -22571 -22262 -21973 -21701 -21+14 -212111 -2OH9 -20749
112 -20S37 -lD315 -201-41 -19954 -1977-4 -19600 -190431 -19268 -19110 -18957
113 -18B08 -18663 -18522 -18384 -1B250 -18119 -17991 -17866 -177 -1762-4
104 -17507 -17392 -17279 -17169 -17060 -16954 -16849 -16747 -16646 -16546
1115 -16449 -16l52 -16258 -16164 -16072 -15982 -15893 -15805 -15718 -15632
1116 -15548 -15464 -15382 -15301 -152211 -151-41 -15063 -1-4 -11909 -104833
117 -14758 -14684 -1461 -1-4538 -1-4466 -14395 -14325 -um -14187 -14118
1M -14051 -1J9IM -13917 -1J8Sl -1J787 -1J722 -13658 -US95 -13532 -13469
119 -Il408 -1J346 -1328S -13225 -1J165 -1J106 -13047 -12_ -12930 -12873
111 -12816 -12759 -Il702 -12646 -12591 -12536 -120481 -12426 -12372 -12319
111 -12265 -12212 -12160 -12107 -12055 -12004 -11952 -11901 -11850 -11800
112 -11751 -11700 -11650 -11601 -11552 -11503 -1165 -11407 -11359 -IIU I
113 -11264 -11217 -11170 -11123 -11077 -1I0U -1D98S -1D939 -10B93 -1111MB
114 -11803 -10758 -lml-4 -IoK9 -1D625 -Iosal -10537 -1G4H -1M50 -111407
115
I117
-10364 -09945 -09542
-1G322 -09904 -D9SQ2
-1079 -09863 -09463
-1G237 -091122 -oMlltt
-101904 -o97B2 -09385
-10152 -09741 -09346
-10110 -09701 -09307
-IJI06 -09661 -09269
-IDOl7 -09amp21 -09230
-09986 -o95BI -09192
118 -09154 -09116 -09078 -09040 -09002 -DlI6S -DB927 -D8B90 -08BS3 -D8B16
119 -08779 -D8742 -D8705 -D8669 -08633 -DBS96 -DI56O -D85lltt -oampt8B -08452
UI -D8416 -DB3BI -0836 -DB310 -0827-4 -08239 -DB21l-4 -DBI69 -18134 -oB099
121 -0B064 -0JI030 -07995 -07961 -D7926 -07892 -07858 -D7B2-f -07790 -01756
122 -07722 -07688 -17655 -07621 -07588 -D7554 -07521 -07_ -07454 -07421
123 -07388 --07356 -07323 -D7290 -D7257 -07225 -07192 -07160 -07128 -07095
124 -07063 -07031 -06999 -06967 -06935 -06903 -D6871 -o6lMO -D6B08 -06776
TABLE AI (Continued)
IIDO 0101 1002 1003 11 OIOS 01116 0007 1001 1009015 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -D6S26 -D6495 -06464 126 -06433 -06403 -06372 -06341 -06311 -062BO -06lS0 -D6219 -06189 -06158 027 -06128 -06098 -06068 -D6038 -06008 -05978 -G5Ha -o591B -05888 -05858 028 -O5B28 -05799 -05769 -D574O -05710 -05681 -05651 -05622 -05592 -05563 029 -0553-4 -05505 -05476 -DS446 -05-417 -05388 -05359 -05330 -05302 -05273 030 -05244 -05215 -05187 -05158 -05129 -05101 -05072 -05044 -05015 -04987 031 -04959 -04930 -04902 -04874 -0-486 -004817 -04789 -04761 -04733 -0-4705 032 -04677 -004649 -04621 -0-4593 -04565 -04538 -04510 -044112 -04454 -Q4427 033 -04399 -04372 -04344 -04316 -04289 -01261 -04234 -0207 -0-4179 -04152 034 -04125 -0-4097 -0-4070 -04043 -0-4016 -03989 -03961 -03934 -03907 -03880 ClE -03853 -03826 -0379 -O3m -03745 -0371 -03691 -Q3665 -03638 -0361 I 111 nlI
-03585 -0331~
-035515 -onn
-0lS3[ -03266
-03505 -03239
-0gt478 -0Z213
-03451 -O pound6
-0H2S -O~IpoundgtO
-03398 -0031gt4
-03372 -01107
-033( -030SI
tlJe -03055 -O302~ -03002 -O2~ipound -021750 -O2ni -02pound -02871 -O2E -02217 ~O9 -O27~3 -027t -027lt 1 -01715 -0261lt -026pound1 -02637 -02GI [ -025pound5 -O25~
IlAO -0253 -02508 -02482 -02~b -02lt10 -O240 -O237a -121S2~ -a22T -O2~O[
1il41 -02275 -OlliO -0222( -02191gt -02173 -021lt7 -02121 -02096 -02070 -02045 112 -02019 -01913 -01968 -01942 -Ol~r7 -0le91 -01866 -0114O -01815 -01789 O4l -01764 -01738 -01713 -01687 -01662 -0637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -01231 -01206 -01181 -01156 -01130 -01105 -0 lOBO -01055 -01030 046 -01004 -00979 -00954 -00929 -0G904 -00878 -00853 -00828 -00B03 -00778 047 -00753 -00728 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0G476 -0G451 -00426 -00401 -00376 -00351 -00326 -00301 -00276 049 -00251 -00226 -00201 -00175 -00150 -00125 -00100 -00075 -00050 -OoOlS 050 00000 OoolS 00050 00075 00100 OOllS 00150 00175 002111 00226 051 00251 00276 00301 00326 00351 00376 00401 0G426 0G451 00476 ts2 00502 00527 00552 00577 00602 0D627 0G6S2 00677 00702 00728 053 00753 00778 00803 0G82B 00853 0G87B 0090-4 00929 0095-4 00979 054 01004 01030 01055 OIOBO 01105 01130 01156 01181 01206 01231
c
I ampII
Tabla AI (Continued)
1- IDDI 1DD2 UDJ D8D4 IDDS 1DD6 U07 ICIOI UDt
155 01257 01282 01307 01332 oll58 oll83 ola 01304 01 59 DIo484 156 01510 01535 01560 01586 01611 0 I 6rT 01662 01687 01711 Dl718 D57 01764 01789 01815 01840 01866 01891 01917 01942 01968 0 I 99l 151 DlOI9 02fM5 02070 112096 02111 021-47 02173 02198 0122 f1225O 059 02215 02l01 02327 O2lSJ 02l71 02-404 02 10 02-456 02C 02508 161 O25ll 0l559 02585 02611 026l7 0266l Dl689 02715 1U71 02767 161 0219) 02819 02845 02B71 D289II 02924 02950 02976 03002 D1029 162 03055 03081 03107 031304 03160 03186 0311l 0J2l9 0l266 D32n 161 03319 033045 O3m 03398 030415 03451 U47I 03505 o3SlI Dl558 D6ltt 0358S 03611 03638 03665 0369l 03719 03745 urn cum 03826 us 03853 038110 03907 039304 03961 03989 0-4016 0-4043 0-4070 0-4097 G66 025 0151 0-4179 0 bull Q07 0-41304 0161 0219 0-4316 O l U371 D67 Ul99 044l7 0 5-4 0+182 04510 0 538 0565 04593 011 0-4649 I6D 04677 0705 0m3 0-4761 0789 017 0-48-45 0-487 0 901 0910 169 0-4959 0987 osol5 oso+f O507l 05101 051l9 05158 05187 05115 17D 05l+i D5l73 D5302 05310 05359 05_ oHI7 05+t6 05-476 ossos D71 055304 05563 05592 D56ll 05651 05681 D5710 D57-40 05769 05799 In 058l8 05858 05888 05918 059-48 05978 D6OO8 06038 D6068 D6098 Dn 06128 06158 06189 06l19 II6l5O 06280 06311 063041 DU71 D6403 17-4 06-433 06-464 060495 G65l6 06557 06588 D6QD 06amp51 0668l 06713 US 067 5 06776 06808 D684O 06871 06903 06935 06967 06999 07031 176 07063 07095 07128 07160 07192 07l25 onsJ Ono 07311 07356 D77 07388 0721 0745-4 OJ D7511 Oj55-4 07588 07611 07655 07688 D7I 07711 07756 07790 07114 07858 071192 079l6 07961 07995 08010 D 011064 08099 0813 08169 08204 08l3J 08l74 08310 0835 08l81 DM 08416 08452 0 oas24 08560 oas96 08633 08669 08705 D87-42 111 08779 08816 08853 08890 D8917 08965 09002 090-40 09071 09116 Da2 0915-4 09192 G92lO 09169 09307 09346 G9385 09-424 09463 09502
Table AI (ContInued)
DG IoDI DoD2 DDOl D LIDS DDD6 DDD7 1001 11D9
Da3 095-42 09581 09621 09661 G9701 09741 09782 09822 09863 09904 DoN 09945 09986 10027 1D069 10110 10152 1019gt4 10237 10279 10322 DaS 10364 10407 10450 1049gt4 10537 IOS81 1D6l5 10669 1071 10758 186 10803 10848 10893 10939 10985 11031 11077 11123 11170 11217 0B1 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 lUi 11750 lI80() Ll85() 11 9() I 1I~52 12004 12055 12107 2160 12212 alll~middot pound2265 IUIS [nn 12426 L24S1 2S36 f25~[ 12~6 [ 2702 L175~
QIQ 12Bt6 12Em i2nD L29BB t3M7 1310( 13165 13l25 32E5 131-~C
SS[ IHO~ ~E~ [m poundi57$ 365C ln2 117CT r 3eS2 1717 f7l rt
~~2 [4Q[1 I~I( lmiddotmmiddotT flt2ES IQI f bull tJpound 1ltlt66 Ilt~ [A~II i 468lt U [i7S [4823 [lt9(1~ [Age~ S(l(~ 15[lt[ 1522( [5301 (36 ~4H Qbull( 155ltamp [51gt32 [5718 15005 15e5~ 19gt8 [(072 16[(( [6258 [63S2 Ui L~49 16S46 LeuroG46 167lt7 1(849 [695-1 17060 l7169 17279 73n 4) 17507 176l4 7741l 17866 17991 1amp1 9 18250 1pound384 [8522 1l63 cn 18808 1B157 19110 19268 19431 100 1977-4 19954 20141 20335 098 20537 20749 20969 21201 21444 21701 21973 22262 22571 22904 U9 23263 23656 2-4089 24573 25121 25758 26521 2778 28782 30902
SouRCE Generated by IL L man Used with permission
bull The emries In this table are quamiIes ~ of the standard normal random YIlIiabIe Z eIeted iO P(Z s tJ - and P(Z gt tJ - I - p Note that the value of co two decimal pIacas determines which raw co use the third decimal pIKe of Pdetermines which column to use to find Zp
ampII
=
510 APPeNDIX APPENDIX 511
TABLEAl Chl-Squd Dlatrlbutlonmiddot TABLEA3 Binomial DIstribution
0750 otoO 0950 0975 0990 ot95 9 n I 005 010 015 01pound1 elgt 010 015 00 05
k =I 1 1 5 6 7 8 9
10 II 11 13 15 16 17 18 19 10 11 11 11 1 15 16 17 11 19 10 ltto SO 60 70 80
100 Z
UU 1773 lt4108 5385 6626 78011 9037
10n 1139 1155 1370 1lt485 1598 1711 IUS 1937 10lt49 1160 lln 1381 193 1604 27 281 291 303 3153 3262 3371 380 62 5633 amp698 7758 8813 9865
lot1 0675
1706 605 6251 7779 9136
10M 1101 1116 1lt468 1599 1718 1855 1981 1106 nll 235lt4 1lt477 1599 1710 18lt41 1961 lUI 3101 3UO 318 3556 367lt4 3791 39ot lt40l6 5181 6311 7lt40 8553 9658
1076 1185
281
31111 5991 7815 9lt188
1107 1159 107 1551 1691 1831 1968 1103 1136 2368 1500 lUO 1759 2887 301 31 1 3167 1191 3517 36lt42 3765 3889 lt4011 1301 4156 377 5576 6750 7908 to53
IOU 1131 1143
16lt15
501 7378 9]f8
111 1l8l 1 5 1601 1753 1902 lM8 1191 23]f 17 l6ll 17lt49 1885 1019 3153 3185 3417 354 3618 3808 1917 lt4065 lt4192 119 lt4sn 698 593 71lt41 8330 9502
1066 1181 1296
1960
6635 9110
113 1328 1501 1681 184 1009 1167 lUI 1lt473 16ll 1769 1 3058 1100 33lt11 11 3619 3757 lB93 lt4029 16lt1
~8 31 lt4564 -96 ltI8l8 959 SU 6369 1615 8838
10M 1111 111 1358
1326
717 1060 1114 16 1675 1855 lUI 113n 1519 2676 1830 111 31]1 ]110 1lt417 nn 1716 3158 -4000 I 110 1 556 -93 4l9
4so 5134 5367 6677 19lt4 195
1042 IIU 1183 10101
lS76
1083 1381 1617 18017 1051
n 1lt431 1611 1788 195 lr16 1191 3Ul 1611 3770 315 lt4079 131 381 4532 lt4410 17 973 5118 5161 05 55 561 5130 5970 73 IU6 9961
1111 128 1371 14
3090
1
3
5
6
7
0
0 I 1
0
1 1
0 I 1 1 lt4
0 I 1 1 4 5
0 I 1 1 5 6
0 I 1 3 4
09500 10000
09025 09975 10000
0857lt4 09928 09999 10000
081lt45 09860 09995 10000 10000
07738 09714 09988 10000 10000 10000
07351 096n 09918 09999
10000 10000 10000
06983 09556 09962 09998 10000
09000 10000
08100 09900 10000
0n9O uno 09990 10000
06561 09477 09963 09999 10000
05905 09185 0991lt4 09995 10000 10000
0531 Q8857 09842 09981 09999 10000 10000
0lt4183 08501 097lt43 09973 09998
08500 10000
07225 09775 10000
06141 09392 09966 10000
05220 08905 09880 09995 10000
04431 08352 09134 09918 09999 10000
03171 01165 09527 09941 09996 10000 10000
03206 01166 09262 09879 09988
OBOOO 10000
06400 000 10000
OSlili
00960 09910 10000
010) OBI92 O912(J 099114 10000
0317 07373 09411 099D 09997 10000
02611 0655[ 09011 09830 09984 O99) 10000
01097 05767 03510 09667 09953
llSOO 10000
t)5~25
o~37S
1000u (H21~
oll43a 09841 10000
021 161 omlJ 0I92 091 10000
on73 00)28 OflS 09044 09990 10000
o171l0 05339 011306 014 09954 099WI 10000
01335 D4middot-H9 07564 O92N 09071
07000 10000
04900 09100 10000
03430 07840 09730 10000
02401 06511 09163 09919 10000
01681 052B2 08369 09692 09916 10000
01116 0A202 01443 09295 09B91 09993 10000
00824 03294 06411 08140 09712
06500 10000
04225 08775 10000
02146 01182 09511 10000
01185 05630 08735 09850 10000
01160 U284 01648 09460 09941 10000
00154 01191 06411 08826 09117 09982 10000
00490 02338 05323 08002 09444
Q6000 10000
03600 08400 10000
02160 06480 09360 10000
01296 0lt4152 08208 09144 10000
00778 01310 06826 09130 09898 10000
00461 02333 05443 08208 09590 09959 10000
00280 01586 04199 01102 09037
05500 10000
03025 07975 10000
01664 05148 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00277 01636 04415 07447
09308 09911 10000
00152 01024 01164 06083 08471
for gt 100 lUI lba ~ w - (i)(z + Vlk shy 1)1 or tIM monI ~ w -
Ie (I shy ~+ z ft) wh Z II tIM wlua from tIM _danIIud lIOIIIIII dllaiblldaft IIhown In lba bouom
5 6 7
10000 10000 10000
10000 10000 10000
09999 10000 10000
099 10000 10000
09997 09999 10000
09962 09998 10000
09910 09994 10000
09812 09984 10000
09643 09963 10000
of lba IIIbIa ScuIcE Abridpd from Tabla S VoL I of Puraon and Hanlay (191) wllb parmlNlon from tIM ~ Tru_ bullTha IllIriu In lbl tabla ara quantIIeIl of II chi-lIIl11n1d random wrlalH W with du of rndom bullbull IecuMlIO P(W gtIi W) - Pand P(W gt - I - P
512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
II 050 155 060 065 070 075 010 085 G90 095I
0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
I 0999 09917 09661 O91J-l 08343 07297 06089 04826 Ol 10000 09991 09944 0980--1 (9511 09012 08283 07l3-4 062 0 00625 00-410 00256 00150 00081 00039 00016 00005 00001 00000
I 03125 025 01791 01265 00837 00501 00272 00120 00037 00005 5 10000 09999 09994 099 09900 0977 0ff60f 09006 08l22 06875 06090 052 0370 0383 02617 01808 01095 00523 001-40 6 10000 10000 10000 099) 09991 09957 09888 09750 095023 09375 09085 0870-4 01215 07599 06836 059Oltt 0-4780 03-439 01855 7 10000 10000 10000 10000 U999l 09996 09986 09962 09909 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 8 10000 10000 10000 10000 10000 10000 09999 09997 09991
J 0 00312 00185 00102 00053 0002 00010 00003 00001 00000 00000 9 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 01875 01312 00870 005-40 00308 00156 00067 ooon 00005 00000 10 0 05987 03-487 01969 (JI07-J 0OS63 00282 00ll5 00060 00025 2 05000 0-4069 0317 02352 01631 01035 00579 00266 00086 00012 I G9139 07361 05443 03750 u2-l4() 0193 00860 00-464 002l3 3 08125 0738 06630 05716 0718 03672 01627 016-48 00815 0Gl26 1 09885 09298 08202 06778 05156 03828 02616 01673 00996 4 09688 09-497 onn 088-40 08319 01627 06723 05563 0-4095 02262 J 09990 09872 09500 Qrl79 I (J71S) 06496 05138 03823 02660 J 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 09999 0998-4 09901 072 09119 0B497 07515 06lll 050-4-4
6 0 00156 00083 000-41 00018 00007 00G02 00001 00000 00000 00000 5 10000 09999 09986 099) u9il03 09527 09051 083l8 07l8-4 1 OlOff 00692 bull 0G410 oom 00109 00G46 00016 0000-4 00001 00000 6 10000 10000 09999 09991 09965 098 097-4Q 090452 089BO l 0338 02553 01791 0117 00705 00376 0G170 00059 00013 00001 7 10000 10000 10000 0999) 099 09984 09952 09877 09726 3 06562 05585 M557 03529 02557 016904 00989 00-473 00158 00022 8 10000 10000 10000 10000 I()OOO 09999 09995 09983 09955 08906 0836-4 07667 06809 05789 0-4661 U446 onl5 03 00328 9 10000 10000 10000 10000 10000 10000 10000 09999 09997 J 09ampM 09723 09533 0916 0882 uno 07379 06229 04686 026-49 10 10000 10000 10000 00011 10000 10000 10000 10000 10000
6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
6 urn 0980f8 09720 G9510 09176 08665 07903 0679-f 05217 03017 8 10000 10000 10000 10000 09999 0999-4 09980 01 098527 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 10000 10000 10000 09998 0999l 09978
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IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
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3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
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16 17
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1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
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TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
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00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
Page 2
OUESTION THREE [20 marks]
Two judges rated the participants in a dance contest as fol1ows
Contestant
Judge A B C D E F G H
1 5 2 6 3 4 8 7
2 3 1 7 4 5 2 6 8
Are the two rankings independent Use either Spearmans p test or Kendalls T test with (l =005
OUESTION FOUR [20 marks]
Two computer software packages are being considered for use in the inventory control department ofa small manufacturing firm The firm has selected 12 different computing tasks that are typical of the kinds ofjobs such a package would have to perform and then recorded tle number of seconds each package required to complete each task given in the following table Do the data present sufficient evidence to indicate that both softwares performed the jobs at the same time Analyse the data by using the Wilcoxon Signed Rank Test with (l = 010
Computing Time Required for Software Task Packages X and Y
A B C D E F G H I J K L
240 167 216 237 375 314 149 373 179 155 290 199
231 204 177 207 421 361 218 403 260 155 354 255
OUESTION FIVE [20 marks]
A random sample of five sixth-grade boys in one section of town were given a literacy test with the following results 82 74 87 86 75 Another random sample ofeight sixth-grade boys from a different section of town were given the same literacy test with these scores reSUlting 88 77 9188949383 94 Using Smimov test find whether there is a difference in literacy in the two populations of sixthshygrade boys
i
TABLE AI Nonnal Distributionshy
SeJecred wIuu uooI -37190 z- - -32905 ZuIs -19600 Zu -16449
r- 37190 z-shy 32905 ZuIs = 19600 Zushy 16449 tIIIII 1001 1002 10D3 1104 1005 11116 l1li7 1111 1119
IDD -3D9Ol -2I7Bl -27478 -265l1 -25758 -25121 -24573 -2laquo189 -2J656
111 -23263 -U904 -22571 -22262 -21973 -21701 -21+14 -212111 -2OH9 -20749
112 -20S37 -lD315 -201-41 -19954 -1977-4 -19600 -190431 -19268 -19110 -18957
113 -18B08 -18663 -18522 -18384 -1B250 -18119 -17991 -17866 -177 -1762-4
104 -17507 -17392 -17279 -17169 -17060 -16954 -16849 -16747 -16646 -16546
1115 -16449 -16l52 -16258 -16164 -16072 -15982 -15893 -15805 -15718 -15632
1116 -15548 -15464 -15382 -15301 -152211 -151-41 -15063 -1-4 -11909 -104833
117 -14758 -14684 -1461 -1-4538 -1-4466 -14395 -14325 -um -14187 -14118
1M -14051 -1J9IM -13917 -1J8Sl -1J787 -1J722 -13658 -US95 -13532 -13469
119 -Il408 -1J346 -1328S -13225 -1J165 -1J106 -13047 -12_ -12930 -12873
111 -12816 -12759 -Il702 -12646 -12591 -12536 -120481 -12426 -12372 -12319
111 -12265 -12212 -12160 -12107 -12055 -12004 -11952 -11901 -11850 -11800
112 -11751 -11700 -11650 -11601 -11552 -11503 -1165 -11407 -11359 -IIU I
113 -11264 -11217 -11170 -11123 -11077 -1I0U -1D98S -1D939 -10B93 -1111MB
114 -11803 -10758 -lml-4 -IoK9 -1D625 -Iosal -10537 -1G4H -1M50 -111407
115
I117
-10364 -09945 -09542
-1G322 -09904 -D9SQ2
-1079 -09863 -09463
-1G237 -091122 -oMlltt
-101904 -o97B2 -09385
-10152 -09741 -09346
-10110 -09701 -09307
-IJI06 -09661 -09269
-IDOl7 -09amp21 -09230
-09986 -o95BI -09192
118 -09154 -09116 -09078 -09040 -09002 -DlI6S -DB927 -D8B90 -08BS3 -D8B16
119 -08779 -D8742 -D8705 -D8669 -08633 -DBS96 -DI56O -D85lltt -oampt8B -08452
UI -D8416 -DB3BI -0836 -DB310 -0827-4 -08239 -DB21l-4 -DBI69 -18134 -oB099
121 -0B064 -0JI030 -07995 -07961 -D7926 -07892 -07858 -D7B2-f -07790 -01756
122 -07722 -07688 -17655 -07621 -07588 -D7554 -07521 -07_ -07454 -07421
123 -07388 --07356 -07323 -D7290 -D7257 -07225 -07192 -07160 -07128 -07095
124 -07063 -07031 -06999 -06967 -06935 -06903 -D6871 -o6lMO -D6B08 -06776
TABLE AI (Continued)
IIDO 0101 1002 1003 11 OIOS 01116 0007 1001 1009015 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -D6S26 -D6495 -06464 126 -06433 -06403 -06372 -06341 -06311 -062BO -06lS0 -D6219 -06189 -06158 027 -06128 -06098 -06068 -D6038 -06008 -05978 -G5Ha -o591B -05888 -05858 028 -O5B28 -05799 -05769 -D574O -05710 -05681 -05651 -05622 -05592 -05563 029 -0553-4 -05505 -05476 -DS446 -05-417 -05388 -05359 -05330 -05302 -05273 030 -05244 -05215 -05187 -05158 -05129 -05101 -05072 -05044 -05015 -04987 031 -04959 -04930 -04902 -04874 -0-486 -004817 -04789 -04761 -04733 -0-4705 032 -04677 -004649 -04621 -0-4593 -04565 -04538 -04510 -044112 -04454 -Q4427 033 -04399 -04372 -04344 -04316 -04289 -01261 -04234 -0207 -0-4179 -04152 034 -04125 -0-4097 -0-4070 -04043 -0-4016 -03989 -03961 -03934 -03907 -03880 ClE -03853 -03826 -0379 -O3m -03745 -0371 -03691 -Q3665 -03638 -0361 I 111 nlI
-03585 -0331~
-035515 -onn
-0lS3[ -03266
-03505 -03239
-0gt478 -0Z213
-03451 -O pound6
-0H2S -O~IpoundgtO
-03398 -0031gt4
-03372 -01107
-033( -030SI
tlJe -03055 -O302~ -03002 -O2~ipound -021750 -O2ni -02pound -02871 -O2E -02217 ~O9 -O27~3 -027t -027lt 1 -01715 -0261lt -026pound1 -02637 -02GI [ -025pound5 -O25~
IlAO -0253 -02508 -02482 -02~b -02lt10 -O240 -O237a -121S2~ -a22T -O2~O[
1il41 -02275 -OlliO -0222( -02191gt -02173 -021lt7 -02121 -02096 -02070 -02045 112 -02019 -01913 -01968 -01942 -Ol~r7 -0le91 -01866 -0114O -01815 -01789 O4l -01764 -01738 -01713 -01687 -01662 -0637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -01231 -01206 -01181 -01156 -01130 -01105 -0 lOBO -01055 -01030 046 -01004 -00979 -00954 -00929 -0G904 -00878 -00853 -00828 -00B03 -00778 047 -00753 -00728 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0G476 -0G451 -00426 -00401 -00376 -00351 -00326 -00301 -00276 049 -00251 -00226 -00201 -00175 -00150 -00125 -00100 -00075 -00050 -OoOlS 050 00000 OoolS 00050 00075 00100 OOllS 00150 00175 002111 00226 051 00251 00276 00301 00326 00351 00376 00401 0G426 0G451 00476 ts2 00502 00527 00552 00577 00602 0D627 0G6S2 00677 00702 00728 053 00753 00778 00803 0G82B 00853 0G87B 0090-4 00929 0095-4 00979 054 01004 01030 01055 OIOBO 01105 01130 01156 01181 01206 01231
c
I ampII
Tabla AI (Continued)
1- IDDI 1DD2 UDJ D8D4 IDDS 1DD6 U07 ICIOI UDt
155 01257 01282 01307 01332 oll58 oll83 ola 01304 01 59 DIo484 156 01510 01535 01560 01586 01611 0 I 6rT 01662 01687 01711 Dl718 D57 01764 01789 01815 01840 01866 01891 01917 01942 01968 0 I 99l 151 DlOI9 02fM5 02070 112096 02111 021-47 02173 02198 0122 f1225O 059 02215 02l01 02327 O2lSJ 02l71 02-404 02 10 02-456 02C 02508 161 O25ll 0l559 02585 02611 026l7 0266l Dl689 02715 1U71 02767 161 0219) 02819 02845 02B71 D289II 02924 02950 02976 03002 D1029 162 03055 03081 03107 031304 03160 03186 0311l 0J2l9 0l266 D32n 161 03319 033045 O3m 03398 030415 03451 U47I 03505 o3SlI Dl558 D6ltt 0358S 03611 03638 03665 0369l 03719 03745 urn cum 03826 us 03853 038110 03907 039304 03961 03989 0-4016 0-4043 0-4070 0-4097 G66 025 0151 0-4179 0 bull Q07 0-41304 0161 0219 0-4316 O l U371 D67 Ul99 044l7 0 5-4 0+182 04510 0 538 0565 04593 011 0-4649 I6D 04677 0705 0m3 0-4761 0789 017 0-48-45 0-487 0 901 0910 169 0-4959 0987 osol5 oso+f O507l 05101 051l9 05158 05187 05115 17D 05l+i D5l73 D5302 05310 05359 05_ oHI7 05+t6 05-476 ossos D71 055304 05563 05592 D56ll 05651 05681 D5710 D57-40 05769 05799 In 058l8 05858 05888 05918 059-48 05978 D6OO8 06038 D6068 D6098 Dn 06128 06158 06189 06l19 II6l5O 06280 06311 063041 DU71 D6403 17-4 06-433 06-464 060495 G65l6 06557 06588 D6QD 06amp51 0668l 06713 US 067 5 06776 06808 D684O 06871 06903 06935 06967 06999 07031 176 07063 07095 07128 07160 07192 07l25 onsJ Ono 07311 07356 D77 07388 0721 0745-4 OJ D7511 Oj55-4 07588 07611 07655 07688 D7I 07711 07756 07790 07114 07858 071192 079l6 07961 07995 08010 D 011064 08099 0813 08169 08204 08l3J 08l74 08310 0835 08l81 DM 08416 08452 0 oas24 08560 oas96 08633 08669 08705 D87-42 111 08779 08816 08853 08890 D8917 08965 09002 090-40 09071 09116 Da2 0915-4 09192 G92lO 09169 09307 09346 G9385 09-424 09463 09502
Table AI (ContInued)
DG IoDI DoD2 DDOl D LIDS DDD6 DDD7 1001 11D9
Da3 095-42 09581 09621 09661 G9701 09741 09782 09822 09863 09904 DoN 09945 09986 10027 1D069 10110 10152 1019gt4 10237 10279 10322 DaS 10364 10407 10450 1049gt4 10537 IOS81 1D6l5 10669 1071 10758 186 10803 10848 10893 10939 10985 11031 11077 11123 11170 11217 0B1 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 lUi 11750 lI80() Ll85() 11 9() I 1I~52 12004 12055 12107 2160 12212 alll~middot pound2265 IUIS [nn 12426 L24S1 2S36 f25~[ 12~6 [ 2702 L175~
QIQ 12Bt6 12Em i2nD L29BB t3M7 1310( 13165 13l25 32E5 131-~C
SS[ IHO~ ~E~ [m poundi57$ 365C ln2 117CT r 3eS2 1717 f7l rt
~~2 [4Q[1 I~I( lmiddotmmiddotT flt2ES IQI f bull tJpound 1ltlt66 Ilt~ [A~II i 468lt U [i7S [4823 [lt9(1~ [Age~ S(l(~ 15[lt[ 1522( [5301 (36 ~4H Qbull( 155ltamp [51gt32 [5718 15005 15e5~ 19gt8 [(072 16[(( [6258 [63S2 Ui L~49 16S46 LeuroG46 167lt7 1(849 [695-1 17060 l7169 17279 73n 4) 17507 176l4 7741l 17866 17991 1amp1 9 18250 1pound384 [8522 1l63 cn 18808 1B157 19110 19268 19431 100 1977-4 19954 20141 20335 098 20537 20749 20969 21201 21444 21701 21973 22262 22571 22904 U9 23263 23656 2-4089 24573 25121 25758 26521 2778 28782 30902
SouRCE Generated by IL L man Used with permission
bull The emries In this table are quamiIes ~ of the standard normal random YIlIiabIe Z eIeted iO P(Z s tJ - and P(Z gt tJ - I - p Note that the value of co two decimal pIacas determines which raw co use the third decimal pIKe of Pdetermines which column to use to find Zp
ampII
=
510 APPeNDIX APPENDIX 511
TABLEAl Chl-Squd Dlatrlbutlonmiddot TABLEA3 Binomial DIstribution
0750 otoO 0950 0975 0990 ot95 9 n I 005 010 015 01pound1 elgt 010 015 00 05
k =I 1 1 5 6 7 8 9
10 II 11 13 15 16 17 18 19 10 11 11 11 1 15 16 17 11 19 10 ltto SO 60 70 80
100 Z
UU 1773 lt4108 5385 6626 78011 9037
10n 1139 1155 1370 1lt485 1598 1711 IUS 1937 10lt49 1160 lln 1381 193 1604 27 281 291 303 3153 3262 3371 380 62 5633 amp698 7758 8813 9865
lot1 0675
1706 605 6251 7779 9136
10M 1101 1116 1lt468 1599 1718 1855 1981 1106 nll 235lt4 1lt477 1599 1710 18lt41 1961 lUI 3101 3UO 318 3556 367lt4 3791 39ot lt40l6 5181 6311 7lt40 8553 9658
1076 1185
281
31111 5991 7815 9lt188
1107 1159 107 1551 1691 1831 1968 1103 1136 2368 1500 lUO 1759 2887 301 31 1 3167 1191 3517 36lt42 3765 3889 lt4011 1301 4156 377 5576 6750 7908 to53
IOU 1131 1143
16lt15
501 7378 9]f8
111 1l8l 1 5 1601 1753 1902 lM8 1191 23]f 17 l6ll 17lt49 1885 1019 3153 3185 3417 354 3618 3808 1917 lt4065 lt4192 119 lt4sn 698 593 71lt41 8330 9502
1066 1181 1296
1960
6635 9110
113 1328 1501 1681 184 1009 1167 lUI 1lt473 16ll 1769 1 3058 1100 33lt11 11 3619 3757 lB93 lt4029 16lt1
~8 31 lt4564 -96 ltI8l8 959 SU 6369 1615 8838
10M 1111 111 1358
1326
717 1060 1114 16 1675 1855 lUI 113n 1519 2676 1830 111 31]1 ]110 1lt417 nn 1716 3158 -4000 I 110 1 556 -93 4l9
4so 5134 5367 6677 19lt4 195
1042 IIU 1183 10101
lS76
1083 1381 1617 18017 1051
n 1lt431 1611 1788 195 lr16 1191 3Ul 1611 3770 315 lt4079 131 381 4532 lt4410 17 973 5118 5161 05 55 561 5130 5970 73 IU6 9961
1111 128 1371 14
3090
1
3
5
6
7
0
0 I 1
0
1 1
0 I 1 1 lt4
0 I 1 1 4 5
0 I 1 1 5 6
0 I 1 3 4
09500 10000
09025 09975 10000
0857lt4 09928 09999 10000
081lt45 09860 09995 10000 10000
07738 09714 09988 10000 10000 10000
07351 096n 09918 09999
10000 10000 10000
06983 09556 09962 09998 10000
09000 10000
08100 09900 10000
0n9O uno 09990 10000
06561 09477 09963 09999 10000
05905 09185 0991lt4 09995 10000 10000
0531 Q8857 09842 09981 09999 10000 10000
0lt4183 08501 097lt43 09973 09998
08500 10000
07225 09775 10000
06141 09392 09966 10000
05220 08905 09880 09995 10000
04431 08352 09134 09918 09999 10000
03171 01165 09527 09941 09996 10000 10000
03206 01166 09262 09879 09988
OBOOO 10000
06400 000 10000
OSlili
00960 09910 10000
010) OBI92 O912(J 099114 10000
0317 07373 09411 099D 09997 10000
02611 0655[ 09011 09830 09984 O99) 10000
01097 05767 03510 09667 09953
llSOO 10000
t)5~25
o~37S
1000u (H21~
oll43a 09841 10000
021 161 omlJ 0I92 091 10000
on73 00)28 OflS 09044 09990 10000
o171l0 05339 011306 014 09954 099WI 10000
01335 D4middot-H9 07564 O92N 09071
07000 10000
04900 09100 10000
03430 07840 09730 10000
02401 06511 09163 09919 10000
01681 052B2 08369 09692 09916 10000
01116 0A202 01443 09295 09B91 09993 10000
00824 03294 06411 08140 09712
06500 10000
04225 08775 10000
02146 01182 09511 10000
01185 05630 08735 09850 10000
01160 U284 01648 09460 09941 10000
00154 01191 06411 08826 09117 09982 10000
00490 02338 05323 08002 09444
Q6000 10000
03600 08400 10000
02160 06480 09360 10000
01296 0lt4152 08208 09144 10000
00778 01310 06826 09130 09898 10000
00461 02333 05443 08208 09590 09959 10000
00280 01586 04199 01102 09037
05500 10000
03025 07975 10000
01664 05148 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00277 01636 04415 07447
09308 09911 10000
00152 01024 01164 06083 08471
for gt 100 lUI lba ~ w - (i)(z + Vlk shy 1)1 or tIM monI ~ w -
Ie (I shy ~+ z ft) wh Z II tIM wlua from tIM _danIIud lIOIIIIII dllaiblldaft IIhown In lba bouom
5 6 7
10000 10000 10000
10000 10000 10000
09999 10000 10000
099 10000 10000
09997 09999 10000
09962 09998 10000
09910 09994 10000
09812 09984 10000
09643 09963 10000
of lba IIIbIa ScuIcE Abridpd from Tabla S VoL I of Puraon and Hanlay (191) wllb parmlNlon from tIM ~ Tru_ bullTha IllIriu In lbl tabla ara quantIIeIl of II chi-lIIl11n1d random wrlalH W with du of rndom bullbull IecuMlIO P(W gtIi W) - Pand P(W gt - I - P
512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
II 050 155 060 065 070 075 010 085 G90 095I
0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
I 0999 09917 09661 O91J-l 08343 07297 06089 04826 Ol 10000 09991 09944 0980--1 (9511 09012 08283 07l3-4 062 0 00625 00-410 00256 00150 00081 00039 00016 00005 00001 00000
I 03125 025 01791 01265 00837 00501 00272 00120 00037 00005 5 10000 09999 09994 099 09900 0977 0ff60f 09006 08l22 06875 06090 052 0370 0383 02617 01808 01095 00523 001-40 6 10000 10000 10000 099) 09991 09957 09888 09750 095023 09375 09085 0870-4 01215 07599 06836 059Oltt 0-4780 03-439 01855 7 10000 10000 10000 10000 U999l 09996 09986 09962 09909 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 8 10000 10000 10000 10000 10000 10000 09999 09997 09991
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6 0 00156 00083 000-41 00018 00007 00G02 00001 00000 00000 00000 5 10000 09999 09986 099) u9il03 09527 09051 083l8 07l8-4 1 OlOff 00692 bull 0G410 oom 00109 00G46 00016 0000-4 00001 00000 6 10000 10000 09999 09991 09965 098 097-4Q 090452 089BO l 0338 02553 01791 0117 00705 00376 0G170 00059 00013 00001 7 10000 10000 10000 0999) 099 09984 09952 09877 09726 3 06562 05585 M557 03529 02557 016904 00989 00-473 00158 00022 8 10000 10000 10000 10000 I()OOO 09999 09995 09983 09955 08906 0836-4 07667 06809 05789 0-4661 U446 onl5 03 00328 9 10000 10000 10000 10000 10000 10000 10000 09999 09997 J 09ampM 09723 09533 0916 0882 uno 07379 06229 04686 026-49 10 10000 10000 10000 00011 10000 10000 10000 10000 10000
6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
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IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
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518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
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3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
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10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
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TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
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r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
i
TABLE AI Nonnal Distributionshy
SeJecred wIuu uooI -37190 z- - -32905 ZuIs -19600 Zu -16449
r- 37190 z-shy 32905 ZuIs = 19600 Zushy 16449 tIIIII 1001 1002 10D3 1104 1005 11116 l1li7 1111 1119
IDD -3D9Ol -2I7Bl -27478 -265l1 -25758 -25121 -24573 -2laquo189 -2J656
111 -23263 -U904 -22571 -22262 -21973 -21701 -21+14 -212111 -2OH9 -20749
112 -20S37 -lD315 -201-41 -19954 -1977-4 -19600 -190431 -19268 -19110 -18957
113 -18B08 -18663 -18522 -18384 -1B250 -18119 -17991 -17866 -177 -1762-4
104 -17507 -17392 -17279 -17169 -17060 -16954 -16849 -16747 -16646 -16546
1115 -16449 -16l52 -16258 -16164 -16072 -15982 -15893 -15805 -15718 -15632
1116 -15548 -15464 -15382 -15301 -152211 -151-41 -15063 -1-4 -11909 -104833
117 -14758 -14684 -1461 -1-4538 -1-4466 -14395 -14325 -um -14187 -14118
1M -14051 -1J9IM -13917 -1J8Sl -1J787 -1J722 -13658 -US95 -13532 -13469
119 -Il408 -1J346 -1328S -13225 -1J165 -1J106 -13047 -12_ -12930 -12873
111 -12816 -12759 -Il702 -12646 -12591 -12536 -120481 -12426 -12372 -12319
111 -12265 -12212 -12160 -12107 -12055 -12004 -11952 -11901 -11850 -11800
112 -11751 -11700 -11650 -11601 -11552 -11503 -1165 -11407 -11359 -IIU I
113 -11264 -11217 -11170 -11123 -11077 -1I0U -1D98S -1D939 -10B93 -1111MB
114 -11803 -10758 -lml-4 -IoK9 -1D625 -Iosal -10537 -1G4H -1M50 -111407
115
I117
-10364 -09945 -09542
-1G322 -09904 -D9SQ2
-1079 -09863 -09463
-1G237 -091122 -oMlltt
-101904 -o97B2 -09385
-10152 -09741 -09346
-10110 -09701 -09307
-IJI06 -09661 -09269
-IDOl7 -09amp21 -09230
-09986 -o95BI -09192
118 -09154 -09116 -09078 -09040 -09002 -DlI6S -DB927 -D8B90 -08BS3 -D8B16
119 -08779 -D8742 -D8705 -D8669 -08633 -DBS96 -DI56O -D85lltt -oampt8B -08452
UI -D8416 -DB3BI -0836 -DB310 -0827-4 -08239 -DB21l-4 -DBI69 -18134 -oB099
121 -0B064 -0JI030 -07995 -07961 -D7926 -07892 -07858 -D7B2-f -07790 -01756
122 -07722 -07688 -17655 -07621 -07588 -D7554 -07521 -07_ -07454 -07421
123 -07388 --07356 -07323 -D7290 -D7257 -07225 -07192 -07160 -07128 -07095
124 -07063 -07031 -06999 -06967 -06935 -06903 -D6871 -o6lMO -D6B08 -06776
TABLE AI (Continued)
IIDO 0101 1002 1003 11 OIOS 01116 0007 1001 1009015 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -D6S26 -D6495 -06464 126 -06433 -06403 -06372 -06341 -06311 -062BO -06lS0 -D6219 -06189 -06158 027 -06128 -06098 -06068 -D6038 -06008 -05978 -G5Ha -o591B -05888 -05858 028 -O5B28 -05799 -05769 -D574O -05710 -05681 -05651 -05622 -05592 -05563 029 -0553-4 -05505 -05476 -DS446 -05-417 -05388 -05359 -05330 -05302 -05273 030 -05244 -05215 -05187 -05158 -05129 -05101 -05072 -05044 -05015 -04987 031 -04959 -04930 -04902 -04874 -0-486 -004817 -04789 -04761 -04733 -0-4705 032 -04677 -004649 -04621 -0-4593 -04565 -04538 -04510 -044112 -04454 -Q4427 033 -04399 -04372 -04344 -04316 -04289 -01261 -04234 -0207 -0-4179 -04152 034 -04125 -0-4097 -0-4070 -04043 -0-4016 -03989 -03961 -03934 -03907 -03880 ClE -03853 -03826 -0379 -O3m -03745 -0371 -03691 -Q3665 -03638 -0361 I 111 nlI
-03585 -0331~
-035515 -onn
-0lS3[ -03266
-03505 -03239
-0gt478 -0Z213
-03451 -O pound6
-0H2S -O~IpoundgtO
-03398 -0031gt4
-03372 -01107
-033( -030SI
tlJe -03055 -O302~ -03002 -O2~ipound -021750 -O2ni -02pound -02871 -O2E -02217 ~O9 -O27~3 -027t -027lt 1 -01715 -0261lt -026pound1 -02637 -02GI [ -025pound5 -O25~
IlAO -0253 -02508 -02482 -02~b -02lt10 -O240 -O237a -121S2~ -a22T -O2~O[
1il41 -02275 -OlliO -0222( -02191gt -02173 -021lt7 -02121 -02096 -02070 -02045 112 -02019 -01913 -01968 -01942 -Ol~r7 -0le91 -01866 -0114O -01815 -01789 O4l -01764 -01738 -01713 -01687 -01662 -0637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -01231 -01206 -01181 -01156 -01130 -01105 -0 lOBO -01055 -01030 046 -01004 -00979 -00954 -00929 -0G904 -00878 -00853 -00828 -00B03 -00778 047 -00753 -00728 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0G476 -0G451 -00426 -00401 -00376 -00351 -00326 -00301 -00276 049 -00251 -00226 -00201 -00175 -00150 -00125 -00100 -00075 -00050 -OoOlS 050 00000 OoolS 00050 00075 00100 OOllS 00150 00175 002111 00226 051 00251 00276 00301 00326 00351 00376 00401 0G426 0G451 00476 ts2 00502 00527 00552 00577 00602 0D627 0G6S2 00677 00702 00728 053 00753 00778 00803 0G82B 00853 0G87B 0090-4 00929 0095-4 00979 054 01004 01030 01055 OIOBO 01105 01130 01156 01181 01206 01231
c
I ampII
Tabla AI (Continued)
1- IDDI 1DD2 UDJ D8D4 IDDS 1DD6 U07 ICIOI UDt
155 01257 01282 01307 01332 oll58 oll83 ola 01304 01 59 DIo484 156 01510 01535 01560 01586 01611 0 I 6rT 01662 01687 01711 Dl718 D57 01764 01789 01815 01840 01866 01891 01917 01942 01968 0 I 99l 151 DlOI9 02fM5 02070 112096 02111 021-47 02173 02198 0122 f1225O 059 02215 02l01 02327 O2lSJ 02l71 02-404 02 10 02-456 02C 02508 161 O25ll 0l559 02585 02611 026l7 0266l Dl689 02715 1U71 02767 161 0219) 02819 02845 02B71 D289II 02924 02950 02976 03002 D1029 162 03055 03081 03107 031304 03160 03186 0311l 0J2l9 0l266 D32n 161 03319 033045 O3m 03398 030415 03451 U47I 03505 o3SlI Dl558 D6ltt 0358S 03611 03638 03665 0369l 03719 03745 urn cum 03826 us 03853 038110 03907 039304 03961 03989 0-4016 0-4043 0-4070 0-4097 G66 025 0151 0-4179 0 bull Q07 0-41304 0161 0219 0-4316 O l U371 D67 Ul99 044l7 0 5-4 0+182 04510 0 538 0565 04593 011 0-4649 I6D 04677 0705 0m3 0-4761 0789 017 0-48-45 0-487 0 901 0910 169 0-4959 0987 osol5 oso+f O507l 05101 051l9 05158 05187 05115 17D 05l+i D5l73 D5302 05310 05359 05_ oHI7 05+t6 05-476 ossos D71 055304 05563 05592 D56ll 05651 05681 D5710 D57-40 05769 05799 In 058l8 05858 05888 05918 059-48 05978 D6OO8 06038 D6068 D6098 Dn 06128 06158 06189 06l19 II6l5O 06280 06311 063041 DU71 D6403 17-4 06-433 06-464 060495 G65l6 06557 06588 D6QD 06amp51 0668l 06713 US 067 5 06776 06808 D684O 06871 06903 06935 06967 06999 07031 176 07063 07095 07128 07160 07192 07l25 onsJ Ono 07311 07356 D77 07388 0721 0745-4 OJ D7511 Oj55-4 07588 07611 07655 07688 D7I 07711 07756 07790 07114 07858 071192 079l6 07961 07995 08010 D 011064 08099 0813 08169 08204 08l3J 08l74 08310 0835 08l81 DM 08416 08452 0 oas24 08560 oas96 08633 08669 08705 D87-42 111 08779 08816 08853 08890 D8917 08965 09002 090-40 09071 09116 Da2 0915-4 09192 G92lO 09169 09307 09346 G9385 09-424 09463 09502
Table AI (ContInued)
DG IoDI DoD2 DDOl D LIDS DDD6 DDD7 1001 11D9
Da3 095-42 09581 09621 09661 G9701 09741 09782 09822 09863 09904 DoN 09945 09986 10027 1D069 10110 10152 1019gt4 10237 10279 10322 DaS 10364 10407 10450 1049gt4 10537 IOS81 1D6l5 10669 1071 10758 186 10803 10848 10893 10939 10985 11031 11077 11123 11170 11217 0B1 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 lUi 11750 lI80() Ll85() 11 9() I 1I~52 12004 12055 12107 2160 12212 alll~middot pound2265 IUIS [nn 12426 L24S1 2S36 f25~[ 12~6 [ 2702 L175~
QIQ 12Bt6 12Em i2nD L29BB t3M7 1310( 13165 13l25 32E5 131-~C
SS[ IHO~ ~E~ [m poundi57$ 365C ln2 117CT r 3eS2 1717 f7l rt
~~2 [4Q[1 I~I( lmiddotmmiddotT flt2ES IQI f bull tJpound 1ltlt66 Ilt~ [A~II i 468lt U [i7S [4823 [lt9(1~ [Age~ S(l(~ 15[lt[ 1522( [5301 (36 ~4H Qbull( 155ltamp [51gt32 [5718 15005 15e5~ 19gt8 [(072 16[(( [6258 [63S2 Ui L~49 16S46 LeuroG46 167lt7 1(849 [695-1 17060 l7169 17279 73n 4) 17507 176l4 7741l 17866 17991 1amp1 9 18250 1pound384 [8522 1l63 cn 18808 1B157 19110 19268 19431 100 1977-4 19954 20141 20335 098 20537 20749 20969 21201 21444 21701 21973 22262 22571 22904 U9 23263 23656 2-4089 24573 25121 25758 26521 2778 28782 30902
SouRCE Generated by IL L man Used with permission
bull The emries In this table are quamiIes ~ of the standard normal random YIlIiabIe Z eIeted iO P(Z s tJ - and P(Z gt tJ - I - p Note that the value of co two decimal pIacas determines which raw co use the third decimal pIKe of Pdetermines which column to use to find Zp
ampII
=
510 APPeNDIX APPENDIX 511
TABLEAl Chl-Squd Dlatrlbutlonmiddot TABLEA3 Binomial DIstribution
0750 otoO 0950 0975 0990 ot95 9 n I 005 010 015 01pound1 elgt 010 015 00 05
k =I 1 1 5 6 7 8 9
10 II 11 13 15 16 17 18 19 10 11 11 11 1 15 16 17 11 19 10 ltto SO 60 70 80
100 Z
UU 1773 lt4108 5385 6626 78011 9037
10n 1139 1155 1370 1lt485 1598 1711 IUS 1937 10lt49 1160 lln 1381 193 1604 27 281 291 303 3153 3262 3371 380 62 5633 amp698 7758 8813 9865
lot1 0675
1706 605 6251 7779 9136
10M 1101 1116 1lt468 1599 1718 1855 1981 1106 nll 235lt4 1lt477 1599 1710 18lt41 1961 lUI 3101 3UO 318 3556 367lt4 3791 39ot lt40l6 5181 6311 7lt40 8553 9658
1076 1185
281
31111 5991 7815 9lt188
1107 1159 107 1551 1691 1831 1968 1103 1136 2368 1500 lUO 1759 2887 301 31 1 3167 1191 3517 36lt42 3765 3889 lt4011 1301 4156 377 5576 6750 7908 to53
IOU 1131 1143
16lt15
501 7378 9]f8
111 1l8l 1 5 1601 1753 1902 lM8 1191 23]f 17 l6ll 17lt49 1885 1019 3153 3185 3417 354 3618 3808 1917 lt4065 lt4192 119 lt4sn 698 593 71lt41 8330 9502
1066 1181 1296
1960
6635 9110
113 1328 1501 1681 184 1009 1167 lUI 1lt473 16ll 1769 1 3058 1100 33lt11 11 3619 3757 lB93 lt4029 16lt1
~8 31 lt4564 -96 ltI8l8 959 SU 6369 1615 8838
10M 1111 111 1358
1326
717 1060 1114 16 1675 1855 lUI 113n 1519 2676 1830 111 31]1 ]110 1lt417 nn 1716 3158 -4000 I 110 1 556 -93 4l9
4so 5134 5367 6677 19lt4 195
1042 IIU 1183 10101
lS76
1083 1381 1617 18017 1051
n 1lt431 1611 1788 195 lr16 1191 3Ul 1611 3770 315 lt4079 131 381 4532 lt4410 17 973 5118 5161 05 55 561 5130 5970 73 IU6 9961
1111 128 1371 14
3090
1
3
5
6
7
0
0 I 1
0
1 1
0 I 1 1 lt4
0 I 1 1 4 5
0 I 1 1 5 6
0 I 1 3 4
09500 10000
09025 09975 10000
0857lt4 09928 09999 10000
081lt45 09860 09995 10000 10000
07738 09714 09988 10000 10000 10000
07351 096n 09918 09999
10000 10000 10000
06983 09556 09962 09998 10000
09000 10000
08100 09900 10000
0n9O uno 09990 10000
06561 09477 09963 09999 10000
05905 09185 0991lt4 09995 10000 10000
0531 Q8857 09842 09981 09999 10000 10000
0lt4183 08501 097lt43 09973 09998
08500 10000
07225 09775 10000
06141 09392 09966 10000
05220 08905 09880 09995 10000
04431 08352 09134 09918 09999 10000
03171 01165 09527 09941 09996 10000 10000
03206 01166 09262 09879 09988
OBOOO 10000
06400 000 10000
OSlili
00960 09910 10000
010) OBI92 O912(J 099114 10000
0317 07373 09411 099D 09997 10000
02611 0655[ 09011 09830 09984 O99) 10000
01097 05767 03510 09667 09953
llSOO 10000
t)5~25
o~37S
1000u (H21~
oll43a 09841 10000
021 161 omlJ 0I92 091 10000
on73 00)28 OflS 09044 09990 10000
o171l0 05339 011306 014 09954 099WI 10000
01335 D4middot-H9 07564 O92N 09071
07000 10000
04900 09100 10000
03430 07840 09730 10000
02401 06511 09163 09919 10000
01681 052B2 08369 09692 09916 10000
01116 0A202 01443 09295 09B91 09993 10000
00824 03294 06411 08140 09712
06500 10000
04225 08775 10000
02146 01182 09511 10000
01185 05630 08735 09850 10000
01160 U284 01648 09460 09941 10000
00154 01191 06411 08826 09117 09982 10000
00490 02338 05323 08002 09444
Q6000 10000
03600 08400 10000
02160 06480 09360 10000
01296 0lt4152 08208 09144 10000
00778 01310 06826 09130 09898 10000
00461 02333 05443 08208 09590 09959 10000
00280 01586 04199 01102 09037
05500 10000
03025 07975 10000
01664 05148 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00277 01636 04415 07447
09308 09911 10000
00152 01024 01164 06083 08471
for gt 100 lUI lba ~ w - (i)(z + Vlk shy 1)1 or tIM monI ~ w -
Ie (I shy ~+ z ft) wh Z II tIM wlua from tIM _danIIud lIOIIIIII dllaiblldaft IIhown In lba bouom
5 6 7
10000 10000 10000
10000 10000 10000
09999 10000 10000
099 10000 10000
09997 09999 10000
09962 09998 10000
09910 09994 10000
09812 09984 10000
09643 09963 10000
of lba IIIbIa ScuIcE Abridpd from Tabla S VoL I of Puraon and Hanlay (191) wllb parmlNlon from tIM ~ Tru_ bullTha IllIriu In lbl tabla ara quantIIeIl of II chi-lIIl11n1d random wrlalH W with du of rndom bullbull IecuMlIO P(W gtIi W) - Pand P(W gt - I - P
512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
II 050 155 060 065 070 075 010 085 G90 095I
0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
I 0999 09917 09661 O91J-l 08343 07297 06089 04826 Ol 10000 09991 09944 0980--1 (9511 09012 08283 07l3-4 062 0 00625 00-410 00256 00150 00081 00039 00016 00005 00001 00000
I 03125 025 01791 01265 00837 00501 00272 00120 00037 00005 5 10000 09999 09994 099 09900 0977 0ff60f 09006 08l22 06875 06090 052 0370 0383 02617 01808 01095 00523 001-40 6 10000 10000 10000 099) 09991 09957 09888 09750 095023 09375 09085 0870-4 01215 07599 06836 059Oltt 0-4780 03-439 01855 7 10000 10000 10000 10000 U999l 09996 09986 09962 09909 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 8 10000 10000 10000 10000 10000 10000 09999 09997 09991
J 0 00312 00185 00102 00053 0002 00010 00003 00001 00000 00000 9 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 01875 01312 00870 005-40 00308 00156 00067 ooon 00005 00000 10 0 05987 03-487 01969 (JI07-J 0OS63 00282 00ll5 00060 00025 2 05000 0-4069 0317 02352 01631 01035 00579 00266 00086 00012 I G9139 07361 05443 03750 u2-l4() 0193 00860 00-464 002l3 3 08125 0738 06630 05716 0718 03672 01627 016-48 00815 0Gl26 1 09885 09298 08202 06778 05156 03828 02616 01673 00996 4 09688 09-497 onn 088-40 08319 01627 06723 05563 0-4095 02262 J 09990 09872 09500 Qrl79 I (J71S) 06496 05138 03823 02660 J 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 09999 0998-4 09901 072 09119 0B497 07515 06lll 050-4-4
6 0 00156 00083 000-41 00018 00007 00G02 00001 00000 00000 00000 5 10000 09999 09986 099) u9il03 09527 09051 083l8 07l8-4 1 OlOff 00692 bull 0G410 oom 00109 00G46 00016 0000-4 00001 00000 6 10000 10000 09999 09991 09965 098 097-4Q 090452 089BO l 0338 02553 01791 0117 00705 00376 0G170 00059 00013 00001 7 10000 10000 10000 0999) 099 09984 09952 09877 09726 3 06562 05585 M557 03529 02557 016904 00989 00-473 00158 00022 8 10000 10000 10000 10000 I()OOO 09999 09995 09983 09955 08906 0836-4 07667 06809 05789 0-4661 U446 onl5 03 00328 9 10000 10000 10000 10000 10000 10000 10000 09999 09997 J 09ampM 09723 09533 0916 0882 uno 07379 06229 04686 026-49 10 10000 10000 10000 00011 10000 10000 10000 10000 10000
6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
6 urn 0980f8 09720 G9510 09176 08665 07903 0679-f 05217 03017 8 10000 10000 10000 10000 09999 0999-4 09980 01 098527 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 10000 10000 10000 09998 0999l 09978
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IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
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3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
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16 17
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1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
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TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
I ampII
Tabla AI (Continued)
1- IDDI 1DD2 UDJ D8D4 IDDS 1DD6 U07 ICIOI UDt
155 01257 01282 01307 01332 oll58 oll83 ola 01304 01 59 DIo484 156 01510 01535 01560 01586 01611 0 I 6rT 01662 01687 01711 Dl718 D57 01764 01789 01815 01840 01866 01891 01917 01942 01968 0 I 99l 151 DlOI9 02fM5 02070 112096 02111 021-47 02173 02198 0122 f1225O 059 02215 02l01 02327 O2lSJ 02l71 02-404 02 10 02-456 02C 02508 161 O25ll 0l559 02585 02611 026l7 0266l Dl689 02715 1U71 02767 161 0219) 02819 02845 02B71 D289II 02924 02950 02976 03002 D1029 162 03055 03081 03107 031304 03160 03186 0311l 0J2l9 0l266 D32n 161 03319 033045 O3m 03398 030415 03451 U47I 03505 o3SlI Dl558 D6ltt 0358S 03611 03638 03665 0369l 03719 03745 urn cum 03826 us 03853 038110 03907 039304 03961 03989 0-4016 0-4043 0-4070 0-4097 G66 025 0151 0-4179 0 bull Q07 0-41304 0161 0219 0-4316 O l U371 D67 Ul99 044l7 0 5-4 0+182 04510 0 538 0565 04593 011 0-4649 I6D 04677 0705 0m3 0-4761 0789 017 0-48-45 0-487 0 901 0910 169 0-4959 0987 osol5 oso+f O507l 05101 051l9 05158 05187 05115 17D 05l+i D5l73 D5302 05310 05359 05_ oHI7 05+t6 05-476 ossos D71 055304 05563 05592 D56ll 05651 05681 D5710 D57-40 05769 05799 In 058l8 05858 05888 05918 059-48 05978 D6OO8 06038 D6068 D6098 Dn 06128 06158 06189 06l19 II6l5O 06280 06311 063041 DU71 D6403 17-4 06-433 06-464 060495 G65l6 06557 06588 D6QD 06amp51 0668l 06713 US 067 5 06776 06808 D684O 06871 06903 06935 06967 06999 07031 176 07063 07095 07128 07160 07192 07l25 onsJ Ono 07311 07356 D77 07388 0721 0745-4 OJ D7511 Oj55-4 07588 07611 07655 07688 D7I 07711 07756 07790 07114 07858 071192 079l6 07961 07995 08010 D 011064 08099 0813 08169 08204 08l3J 08l74 08310 0835 08l81 DM 08416 08452 0 oas24 08560 oas96 08633 08669 08705 D87-42 111 08779 08816 08853 08890 D8917 08965 09002 090-40 09071 09116 Da2 0915-4 09192 G92lO 09169 09307 09346 G9385 09-424 09463 09502
Table AI (ContInued)
DG IoDI DoD2 DDOl D LIDS DDD6 DDD7 1001 11D9
Da3 095-42 09581 09621 09661 G9701 09741 09782 09822 09863 09904 DoN 09945 09986 10027 1D069 10110 10152 1019gt4 10237 10279 10322 DaS 10364 10407 10450 1049gt4 10537 IOS81 1D6l5 10669 1071 10758 186 10803 10848 10893 10939 10985 11031 11077 11123 11170 11217 0B1 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 lUi 11750 lI80() Ll85() 11 9() I 1I~52 12004 12055 12107 2160 12212 alll~middot pound2265 IUIS [nn 12426 L24S1 2S36 f25~[ 12~6 [ 2702 L175~
QIQ 12Bt6 12Em i2nD L29BB t3M7 1310( 13165 13l25 32E5 131-~C
SS[ IHO~ ~E~ [m poundi57$ 365C ln2 117CT r 3eS2 1717 f7l rt
~~2 [4Q[1 I~I( lmiddotmmiddotT flt2ES IQI f bull tJpound 1ltlt66 Ilt~ [A~II i 468lt U [i7S [4823 [lt9(1~ [Age~ S(l(~ 15[lt[ 1522( [5301 (36 ~4H Qbull( 155ltamp [51gt32 [5718 15005 15e5~ 19gt8 [(072 16[(( [6258 [63S2 Ui L~49 16S46 LeuroG46 167lt7 1(849 [695-1 17060 l7169 17279 73n 4) 17507 176l4 7741l 17866 17991 1amp1 9 18250 1pound384 [8522 1l63 cn 18808 1B157 19110 19268 19431 100 1977-4 19954 20141 20335 098 20537 20749 20969 21201 21444 21701 21973 22262 22571 22904 U9 23263 23656 2-4089 24573 25121 25758 26521 2778 28782 30902
SouRCE Generated by IL L man Used with permission
bull The emries In this table are quamiIes ~ of the standard normal random YIlIiabIe Z eIeted iO P(Z s tJ - and P(Z gt tJ - I - p Note that the value of co two decimal pIacas determines which raw co use the third decimal pIKe of Pdetermines which column to use to find Zp
ampII
=
510 APPeNDIX APPENDIX 511
TABLEAl Chl-Squd Dlatrlbutlonmiddot TABLEA3 Binomial DIstribution
0750 otoO 0950 0975 0990 ot95 9 n I 005 010 015 01pound1 elgt 010 015 00 05
k =I 1 1 5 6 7 8 9
10 II 11 13 15 16 17 18 19 10 11 11 11 1 15 16 17 11 19 10 ltto SO 60 70 80
100 Z
UU 1773 lt4108 5385 6626 78011 9037
10n 1139 1155 1370 1lt485 1598 1711 IUS 1937 10lt49 1160 lln 1381 193 1604 27 281 291 303 3153 3262 3371 380 62 5633 amp698 7758 8813 9865
lot1 0675
1706 605 6251 7779 9136
10M 1101 1116 1lt468 1599 1718 1855 1981 1106 nll 235lt4 1lt477 1599 1710 18lt41 1961 lUI 3101 3UO 318 3556 367lt4 3791 39ot lt40l6 5181 6311 7lt40 8553 9658
1076 1185
281
31111 5991 7815 9lt188
1107 1159 107 1551 1691 1831 1968 1103 1136 2368 1500 lUO 1759 2887 301 31 1 3167 1191 3517 36lt42 3765 3889 lt4011 1301 4156 377 5576 6750 7908 to53
IOU 1131 1143
16lt15
501 7378 9]f8
111 1l8l 1 5 1601 1753 1902 lM8 1191 23]f 17 l6ll 17lt49 1885 1019 3153 3185 3417 354 3618 3808 1917 lt4065 lt4192 119 lt4sn 698 593 71lt41 8330 9502
1066 1181 1296
1960
6635 9110
113 1328 1501 1681 184 1009 1167 lUI 1lt473 16ll 1769 1 3058 1100 33lt11 11 3619 3757 lB93 lt4029 16lt1
~8 31 lt4564 -96 ltI8l8 959 SU 6369 1615 8838
10M 1111 111 1358
1326
717 1060 1114 16 1675 1855 lUI 113n 1519 2676 1830 111 31]1 ]110 1lt417 nn 1716 3158 -4000 I 110 1 556 -93 4l9
4so 5134 5367 6677 19lt4 195
1042 IIU 1183 10101
lS76
1083 1381 1617 18017 1051
n 1lt431 1611 1788 195 lr16 1191 3Ul 1611 3770 315 lt4079 131 381 4532 lt4410 17 973 5118 5161 05 55 561 5130 5970 73 IU6 9961
1111 128 1371 14
3090
1
3
5
6
7
0
0 I 1
0
1 1
0 I 1 1 lt4
0 I 1 1 4 5
0 I 1 1 5 6
0 I 1 3 4
09500 10000
09025 09975 10000
0857lt4 09928 09999 10000
081lt45 09860 09995 10000 10000
07738 09714 09988 10000 10000 10000
07351 096n 09918 09999
10000 10000 10000
06983 09556 09962 09998 10000
09000 10000
08100 09900 10000
0n9O uno 09990 10000
06561 09477 09963 09999 10000
05905 09185 0991lt4 09995 10000 10000
0531 Q8857 09842 09981 09999 10000 10000
0lt4183 08501 097lt43 09973 09998
08500 10000
07225 09775 10000
06141 09392 09966 10000
05220 08905 09880 09995 10000
04431 08352 09134 09918 09999 10000
03171 01165 09527 09941 09996 10000 10000
03206 01166 09262 09879 09988
OBOOO 10000
06400 000 10000
OSlili
00960 09910 10000
010) OBI92 O912(J 099114 10000
0317 07373 09411 099D 09997 10000
02611 0655[ 09011 09830 09984 O99) 10000
01097 05767 03510 09667 09953
llSOO 10000
t)5~25
o~37S
1000u (H21~
oll43a 09841 10000
021 161 omlJ 0I92 091 10000
on73 00)28 OflS 09044 09990 10000
o171l0 05339 011306 014 09954 099WI 10000
01335 D4middot-H9 07564 O92N 09071
07000 10000
04900 09100 10000
03430 07840 09730 10000
02401 06511 09163 09919 10000
01681 052B2 08369 09692 09916 10000
01116 0A202 01443 09295 09B91 09993 10000
00824 03294 06411 08140 09712
06500 10000
04225 08775 10000
02146 01182 09511 10000
01185 05630 08735 09850 10000
01160 U284 01648 09460 09941 10000
00154 01191 06411 08826 09117 09982 10000
00490 02338 05323 08002 09444
Q6000 10000
03600 08400 10000
02160 06480 09360 10000
01296 0lt4152 08208 09144 10000
00778 01310 06826 09130 09898 10000
00461 02333 05443 08208 09590 09959 10000
00280 01586 04199 01102 09037
05500 10000
03025 07975 10000
01664 05148 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00277 01636 04415 07447
09308 09911 10000
00152 01024 01164 06083 08471
for gt 100 lUI lba ~ w - (i)(z + Vlk shy 1)1 or tIM monI ~ w -
Ie (I shy ~+ z ft) wh Z II tIM wlua from tIM _danIIud lIOIIIIII dllaiblldaft IIhown In lba bouom
5 6 7
10000 10000 10000
10000 10000 10000
09999 10000 10000
099 10000 10000
09997 09999 10000
09962 09998 10000
09910 09994 10000
09812 09984 10000
09643 09963 10000
of lba IIIbIa ScuIcE Abridpd from Tabla S VoL I of Puraon and Hanlay (191) wllb parmlNlon from tIM ~ Tru_ bullTha IllIriu In lbl tabla ara quantIIeIl of II chi-lIIl11n1d random wrlalH W with du of rndom bullbull IecuMlIO P(W gtIi W) - Pand P(W gt - I - P
512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
II 050 155 060 065 070 075 010 085 G90 095I
0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
I 0999 09917 09661 O91J-l 08343 07297 06089 04826 Ol 10000 09991 09944 0980--1 (9511 09012 08283 07l3-4 062 0 00625 00-410 00256 00150 00081 00039 00016 00005 00001 00000
I 03125 025 01791 01265 00837 00501 00272 00120 00037 00005 5 10000 09999 09994 099 09900 0977 0ff60f 09006 08l22 06875 06090 052 0370 0383 02617 01808 01095 00523 001-40 6 10000 10000 10000 099) 09991 09957 09888 09750 095023 09375 09085 0870-4 01215 07599 06836 059Oltt 0-4780 03-439 01855 7 10000 10000 10000 10000 U999l 09996 09986 09962 09909 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 8 10000 10000 10000 10000 10000 10000 09999 09997 09991
J 0 00312 00185 00102 00053 0002 00010 00003 00001 00000 00000 9 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 01875 01312 00870 005-40 00308 00156 00067 ooon 00005 00000 10 0 05987 03-487 01969 (JI07-J 0OS63 00282 00ll5 00060 00025 2 05000 0-4069 0317 02352 01631 01035 00579 00266 00086 00012 I G9139 07361 05443 03750 u2-l4() 0193 00860 00-464 002l3 3 08125 0738 06630 05716 0718 03672 01627 016-48 00815 0Gl26 1 09885 09298 08202 06778 05156 03828 02616 01673 00996 4 09688 09-497 onn 088-40 08319 01627 06723 05563 0-4095 02262 J 09990 09872 09500 Qrl79 I (J71S) 06496 05138 03823 02660 J 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 09999 0998-4 09901 072 09119 0B497 07515 06lll 050-4-4
6 0 00156 00083 000-41 00018 00007 00G02 00001 00000 00000 00000 5 10000 09999 09986 099) u9il03 09527 09051 083l8 07l8-4 1 OlOff 00692 bull 0G410 oom 00109 00G46 00016 0000-4 00001 00000 6 10000 10000 09999 09991 09965 098 097-4Q 090452 089BO l 0338 02553 01791 0117 00705 00376 0G170 00059 00013 00001 7 10000 10000 10000 0999) 099 09984 09952 09877 09726 3 06562 05585 M557 03529 02557 016904 00989 00-473 00158 00022 8 10000 10000 10000 10000 I()OOO 09999 09995 09983 09955 08906 0836-4 07667 06809 05789 0-4661 U446 onl5 03 00328 9 10000 10000 10000 10000 10000 10000 10000 09999 09997 J 09ampM 09723 09533 0916 0882 uno 07379 06229 04686 026-49 10 10000 10000 10000 00011 10000 10000 10000 10000 10000
6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
6 urn 0980f8 09720 G9510 09176 08665 07903 0679-f 05217 03017 8 10000 10000 10000 10000 09999 0999-4 09980 01 098527 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 10000 10000 10000 09998 0999l 09978
10 10000 10000 10000 10000 10000 10000 10000 10000 09998 II 10000 10000 10000 10000 10000 10000 10000 10000 10000
w
-0= -0
-ppppppppppp -PPPPPPP~g~
liei~I~HH~ IlstiH~~J 0-wl Wl wtl- o ~c
-OClClClClClOppiii~iii8~~6sect -it IDOit 1~~I~~~e~l~
iiiampsectsectiiilii -=eIi~e~ig~sectsecti- shyCI ~~t tishy
-COClClOpClOOClCl-PClClClClPClClOO1~~~~~-~2sectoIIE~~f~~I~1 t~tl~~CIsect ~f CI III -~ppfegppsectpp-~p~e~~g~secti
1sect~sect2e~~~ IIIIS~e~io--ppppppppppp -pClCCCCCOClCl1~i~=e88sect11 lit~~~~~8111~ tC~ii- ICIt--I
i~~i~i~isectiii ijsect~iiiiiii
bull -0 0
-ppppppp~
ii~~i~i~~ CI middot~ew-CI8~III et~wIa~ ~~If CIshy I
-COClcepcoo1~~eelO2sect~ ii~~~ii~~etJ~~~IIIpound-CD o~
-pppep=ppp-ppppp=pp 11~~-~S~sectsectIi~Ssectsect~~~ t 11
-pCCClClOCClClao OO8~lf~~Il=- ii~~ii~sectw1-1 _ -S - ~
-ppppp=pp-ppep~p=pp
Ills~i~~11 sect~til~~sectI~- wwshy
-ppppppppp -IiE~~sect=P=
I~~~~~~~I I=~~~~~~I
ii5~i~8sectiiibi~i~~sectiIDCD-W ~I- wshy
I III
C In t a
bull n 0bull s Is ampbull 1
ebull bull VI
= bull
bull
bullbull
p
Co W D
o W
116 APP~DIX APPENDIX 517
IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
00005 00037
0000 I OQoII
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
ooqqo 00000
00000 00000
00000 00000 1
07922 09497
04818 07618
02515 05198
(UI1E 030
OOSOI 01637
00193 oon-4
00067 00327
00011 00123
00006 00041
3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
16 17
10000 10000
10000 10000
10000 10000
10000 10000
LOOOCi 10000
10000
10000 10000 10000
10000 10000
10000 10000
1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
shybull o w
liii~iii~isectsectiiiiiiimi~i~i~i~iiiiiiii
iii~iii~~iisectiiiiiiii~i~~i~~iiiiiiiii
III
bull I p
o
o
p
II
~ 1ft
o
o W In
m Z
5lt C
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
510 APPeNDIX APPENDIX 511
TABLEAl Chl-Squd Dlatrlbutlonmiddot TABLEA3 Binomial DIstribution
0750 otoO 0950 0975 0990 ot95 9 n I 005 010 015 01pound1 elgt 010 015 00 05
k =I 1 1 5 6 7 8 9
10 II 11 13 15 16 17 18 19 10 11 11 11 1 15 16 17 11 19 10 ltto SO 60 70 80
100 Z
UU 1773 lt4108 5385 6626 78011 9037
10n 1139 1155 1370 1lt485 1598 1711 IUS 1937 10lt49 1160 lln 1381 193 1604 27 281 291 303 3153 3262 3371 380 62 5633 amp698 7758 8813 9865
lot1 0675
1706 605 6251 7779 9136
10M 1101 1116 1lt468 1599 1718 1855 1981 1106 nll 235lt4 1lt477 1599 1710 18lt41 1961 lUI 3101 3UO 318 3556 367lt4 3791 39ot lt40l6 5181 6311 7lt40 8553 9658
1076 1185
281
31111 5991 7815 9lt188
1107 1159 107 1551 1691 1831 1968 1103 1136 2368 1500 lUO 1759 2887 301 31 1 3167 1191 3517 36lt42 3765 3889 lt4011 1301 4156 377 5576 6750 7908 to53
IOU 1131 1143
16lt15
501 7378 9]f8
111 1l8l 1 5 1601 1753 1902 lM8 1191 23]f 17 l6ll 17lt49 1885 1019 3153 3185 3417 354 3618 3808 1917 lt4065 lt4192 119 lt4sn 698 593 71lt41 8330 9502
1066 1181 1296
1960
6635 9110
113 1328 1501 1681 184 1009 1167 lUI 1lt473 16ll 1769 1 3058 1100 33lt11 11 3619 3757 lB93 lt4029 16lt1
~8 31 lt4564 -96 ltI8l8 959 SU 6369 1615 8838
10M 1111 111 1358
1326
717 1060 1114 16 1675 1855 lUI 113n 1519 2676 1830 111 31]1 ]110 1lt417 nn 1716 3158 -4000 I 110 1 556 -93 4l9
4so 5134 5367 6677 19lt4 195
1042 IIU 1183 10101
lS76
1083 1381 1617 18017 1051
n 1lt431 1611 1788 195 lr16 1191 3Ul 1611 3770 315 lt4079 131 381 4532 lt4410 17 973 5118 5161 05 55 561 5130 5970 73 IU6 9961
1111 128 1371 14
3090
1
3
5
6
7
0
0 I 1
0
1 1
0 I 1 1 lt4
0 I 1 1 4 5
0 I 1 1 5 6
0 I 1 3 4
09500 10000
09025 09975 10000
0857lt4 09928 09999 10000
081lt45 09860 09995 10000 10000
07738 09714 09988 10000 10000 10000
07351 096n 09918 09999
10000 10000 10000
06983 09556 09962 09998 10000
09000 10000
08100 09900 10000
0n9O uno 09990 10000
06561 09477 09963 09999 10000
05905 09185 0991lt4 09995 10000 10000
0531 Q8857 09842 09981 09999 10000 10000
0lt4183 08501 097lt43 09973 09998
08500 10000
07225 09775 10000
06141 09392 09966 10000
05220 08905 09880 09995 10000
04431 08352 09134 09918 09999 10000
03171 01165 09527 09941 09996 10000 10000
03206 01166 09262 09879 09988
OBOOO 10000
06400 000 10000
OSlili
00960 09910 10000
010) OBI92 O912(J 099114 10000
0317 07373 09411 099D 09997 10000
02611 0655[ 09011 09830 09984 O99) 10000
01097 05767 03510 09667 09953
llSOO 10000
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512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
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0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
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6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
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TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
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14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
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518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
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10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
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TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
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r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
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20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
512 APPENDIX APPENDIX 51lbull TABLEAl (Continued)
II I 005 010 015 Ilm ~15 030 035 00 05TABLEAl (Continued)
8 0 0663-4 0305 02725 IIU 01001 00576 00319 00168 OOOIW I 028 08131 06572 05033 0)71 0255] 01691 01064 00632 1 092 09619 08948 07 06785 05518 OA278 0315-4 02201
II 050 155 060 065 070 075 010 085 G90 095I
0 05000 0500 O bull fOGO G35OO 03000 02500 02000 01500 01000 00500 3 09996 09950 09786 09middot131 08M 08059 07064 0591 OA770 I 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 10000 09996 09971 098 09711 09420 089l9 08263 07396
1 0 02500 02025 01600 01125 00900 00625 00ltt00 Oons 00100 OClO25 5 10000 10000 09998 09990 099511 09887 09747 09502 09115 1 07500 06975 06lttOO 05775 0500 0375 G36OO 02775 01900 00975 6 10000 10000 10000 09991 091 09987 0996-4 09915 G9819 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 7 10000 10000 10000 10000 10000 09999 09998 09993 09983
3 0 01250 00911 006-40 00-429 00270 00156 00080 0003 00010 00001 8 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 05000 M252 03520 02818 G2160 01562 010-40 00601 00280 00072 9 0 06302 0387 02ll6 u1342 00151 00404 00207 00101 00046 1 oe750 08336 078-40 0725-4 06570 05781 0880 03859 02710 0126 I 09288 077-48 05995 04362 (DIlO) 01960 01211 00705 OOl85 3 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 2 09916 070 08591 073m lltOOl OA628 01373 02318 0 95
I 0999 09917 09661 O91J-l 08343 07297 06089 04826 Ol 10000 09991 09944 0980--1 (9511 09012 08283 07l3-4 062 0 00625 00-410 00256 00150 00081 00039 00016 00005 00001 00000
I 03125 025 01791 01265 00837 00501 00272 00120 00037 00005 5 10000 09999 09994 099 09900 0977 0ff60f 09006 08l22 06875 06090 052 0370 0383 02617 01808 01095 00523 001-40 6 10000 10000 10000 099) 09991 09957 09888 09750 095023 09375 09085 0870-4 01215 07599 06836 059Oltt 0-4780 03-439 01855 7 10000 10000 10000 10000 U999l 09996 09986 09962 09909 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 8 10000 10000 10000 10000 10000 10000 09999 09997 09991
J 0 00312 00185 00102 00053 0002 00010 00003 00001 00000 00000 9 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 01875 01312 00870 005-40 00308 00156 00067 ooon 00005 00000 10 0 05987 03-487 01969 (JI07-J 0OS63 00282 00ll5 00060 00025 2 05000 0-4069 0317 02352 01631 01035 00579 00266 00086 00012 I G9139 07361 05443 03750 u2-l4() 0193 00860 00-464 002l3 3 08125 0738 06630 05716 0718 03672 01627 016-48 00815 0Gl26 1 09885 09298 08202 06778 05156 03828 02616 01673 00996 4 09688 09-497 onn 088-40 08319 01627 06723 05563 0-4095 02262 J 09990 09872 09500 Qrl79 I (J71S) 06496 05138 03823 02660 J 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 09999 0998-4 09901 072 09119 0B497 07515 06lll 050-4-4
6 0 00156 00083 000-41 00018 00007 00G02 00001 00000 00000 00000 5 10000 09999 09986 099) u9il03 09527 09051 083l8 07l8-4 1 OlOff 00692 bull 0G410 oom 00109 00G46 00016 0000-4 00001 00000 6 10000 10000 09999 09991 09965 098 097-4Q 090452 089BO l 0338 02553 01791 0117 00705 00376 0G170 00059 00013 00001 7 10000 10000 10000 0999) 099 09984 09952 09877 09726 3 06562 05585 M557 03529 02557 016904 00989 00-473 00158 00022 8 10000 10000 10000 10000 I()OOO 09999 09995 09983 09955 08906 0836-4 07667 06809 05789 0-4661 U446 onl5 03 00328 9 10000 10000 10000 10000 10000 10000 10000 09999 09997 J 09ampM 09723 09533 0916 0882 uno 07379 06229 04686 026-49 10 10000 10000 10000 00011 10000 10000 10000 10000 10000
6 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 I 0 05688 03138 01673 00059 00112 00198 00088 00036 00014 I 08981 0697 OA922 03121 uI71 01110 00606 00]02 001197 0 00078 00037 00016 00006 00G02 00001 00000 00000 00000 00000 l 0980f8 0910-4 07788 06174 o~SSl 03127 02001 01189 00652I 00625 00357 00188 00090 00038 00013 GOOO-4 00001 00000 00000 3 G998-4 09815 09306 08389 01133 05696 0256 02963 019111 Gl266 01529 00963 00556 00288 00129 000-47 00012 00002 00000 4 G9999 09972 098-41 094 sB3S4 07897 0668l 05328 039713 05000 03917 02898 01998 01260 00706 0Oll3 00121 00027 00002 5 10000 G9997 09973 09883 051 09218 085l 07535 063ll 0773-4 06836 05801 0-4177 03529 0136 01 80 00738 00257 00038 6 10000 10000 09997 09980 u9924 09784 09-499 D9006 082625 09375 08976 G8of 07662 06706 05551 Mll3 0283-4 01-497 00-4-4-4 7 10000 10000 10000 09998 09939 09957 09878 09707 09390
6 urn 0980f8 09720 G9510 09176 08665 07903 0679-f 05217 03017 8 10000 10000 10000 10000 09999 0999-4 09980 01 098527 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 10000 10000 10000 09998 0999l 09978
10 10000 10000 10000 10000 10000 10000 10000 10000 09998 II 10000 10000 10000 10000 10000 10000 10000 10000 10000
w
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iiiampsectsectiiilii -=eIi~e~ig~sectsecti- shyCI ~~t tishy
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1sect~sect2e~~~ IIIIS~e~io--ppppppppppp -pClCCCCCOClCl1~i~=e88sect11 lit~~~~~8111~ tC~ii- ICIt--I
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-pppep=ppp-ppppp=pp 11~~-~S~sectsectIi~Ssectsect~~~ t 11
-pCCClClOCClClao OO8~lf~~Il=- ii~~ii~sectw1-1 _ -S - ~
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ii5~i~8sectiiibi~i~~sectiIDCD-W ~I- wshy
I III
C In t a
bull n 0bull s Is ampbull 1
ebull bull VI
= bull
bull
bullbull
p
Co W D
o W
116 APP~DIX APPENDIX 517
IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
00005 00037
0000 I OQoII
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
ooqqo 00000
00000 00000
00000 00000 1
07922 09497
04818 07618
02515 05198
(UI1E 030
OOSOI 01637
00193 oon-4
00067 00327
00011 00123
00006 00041
3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
16 17
10000 10000
10000 10000
10000 10000
10000 10000
LOOOCi 10000
10000
10000 10000 10000
10000 10000
10000 10000
1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
shybull o w
liii~iii~isectsectiiiiiiimi~i~i~i~iiiiiiii
iii~iii~~iisectiiiiiiii~i~~i~~iiiiiiiii
III
bull I p
o
o
p
II
~ 1ft
o
o W In
m Z
5lt C
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
w
-0= -0
-ppppppppppp -PPPPPPP~g~
liei~I~HH~ IlstiH~~J 0-wl Wl wtl- o ~c
-OClClClClClOppiii~iii8~~6sect -it IDOit 1~~I~~~e~l~
iiiampsectsectiiilii -=eIi~e~ig~sectsecti- shyCI ~~t tishy
-COClClOpClOOClCl-PClClClClPClClOO1~~~~~-~2sectoIIE~~f~~I~1 t~tl~~CIsect ~f CI III -~ppfegppsectpp-~p~e~~g~secti
1sect~sect2e~~~ IIIIS~e~io--ppppppppppp -pClCCCCCOClCl1~i~=e88sect11 lit~~~~~8111~ tC~ii- ICIt--I
i~~i~i~isectiii ijsect~iiiiiii
bull -0 0
-ppppppp~
ii~~i~i~~ CI middot~ew-CI8~III et~wIa~ ~~If CIshy I
-COClcepcoo1~~eelO2sect~ ii~~~ii~~etJ~~~IIIpound-CD o~
-pppep=ppp-ppppp=pp 11~~-~S~sectsectIi~Ssectsect~~~ t 11
-pCCClClOCClClao OO8~lf~~Il=- ii~~ii~sectw1-1 _ -S - ~
-ppppp=pp-ppep~p=pp
Ills~i~~11 sect~til~~sectI~- wwshy
-ppppppppp -IiE~~sect=P=
I~~~~~~~I I=~~~~~~I
ii5~i~8sectiiibi~i~~sectiIDCD-W ~I- wshy
I III
C In t a
bull n 0bull s Is ampbull 1
ebull bull VI
= bull
bull
bullbull
p
Co W D
o W
116 APP~DIX APPENDIX 517
IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
00005 00037
0000 I OQoII
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
ooqqo 00000
00000 00000
00000 00000 1
07922 09497
04818 07618
02515 05198
(UI1E 030
OOSOI 01637
00193 oon-4
00067 00327
00011 00123
00006 00041
3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
16 17
10000 10000
10000 10000
10000 10000
10000 10000
LOOOCi 10000
10000
10000 10000 10000
10000 10000
10000 10000
1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
shybull o w
liii~iii~isectsectiiiiiiimi~i~i~i~iiiiiiii
iii~iii~~iisectiiiiiiii~i~~i~~iiiiiiiii
III
bull I p
o
o
p
II
~ 1ft
o
o W In
m Z
5lt C
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
116 APP~DIX APPENDIX 517
IrABLEAl (Continued)
II 050 US UO US 070 US 080 US 090 89S
TABLEAl (ContInued)11 0 00002 00001 00000 00000 00000 00000 00000 00000 00000 00000 I 00032 00011 0000] 00001 00000 00000 00000 00000 00000 00000
II OOS 010 015 )J~ Dll OJO 035 040 0452 00193 00079 00028 00008 00002 00000 00000 00000 00000 00000 ] 00730 00356 00151 00056 00017 00004 00001 00000 00000 00000 IS 0 Q46l1 Q2059 00874 oms tWimiddot1 00047 00016 00005 00001 4 01918 01117 00573 00255 00095 00028 0G006 00001 00000 00000 I 08290 05490 03186 nlal OoaOl 00353 00142 00052 00017 5 03871 02607 01581 001116 00386 00143 00039 00007 00001 00000 2 09638 UI59 06042 0398pound1 02)61 01268 00617 00271 00107 6 06118 Q4731 Ul48 01127 01178 00544 00194 00046 00005 00000 1 09945 09444 08227 OW01 OA613 02969 01727 00905 00424 7 08061 06956 05618 04167 02761 01576 00n6 00239 00043 00002 4 09994 09873 09383 Ofl)5u olll6S 05155 01519 02173 01204 8 09170 08655 07747 G6511 05075 03511 OlO54 0G9ll 00256 00021 I 09999 09978 09831 Oi3l~ )8511gt 07116 056-43 OAOl2 01608 9 09807 09579 09166 oB487 07471 06093 04417 01642 bull 01109 00196 6 10000 09997 0996-4 o)BB 0943-4 08689 07548 06098 01522
10 09968 09917 09804 09576 09150 08416 07151 05565 03410 01184 7 10000 10000 09994 09159 091327 09500 08868 07869 065351 09998 09991 09978 09941 09862 09683 09111 08578 07176 04596 8 10000 10000 09999 09991 u99511 Q9818 09578 09050 08182 2 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 10000 10000 10000 O 09911 09963 09876 09661 09231
13 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 10 10000 10000 10000 10000 OJSl9) 09993 09972 09907 09745 I 00017 00005 00001 00000 00000 00000 00000 00000 00000 00000 II 10000 10000 10000 10000 IOllDO 09999 09995 09981 099372 00112 00041 00011 00001 00001 00000 00000 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09999 09997 099891 00461 00101 00078 00025 00007 00001 00000 00000 00000 00000 13 10000 10000 10000 10000 10000 10000 10000 10000 099994 01134 00698 00311 00116 00040 00010 00002 00000 00000 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 10000I 02905 0788 00977 Q0461 00182 00056 00011 0G002 00000 00000
IS 10000 10000 10000 10(100 L()oOO 10000 10000 10000 100006 05000 03561 0l288 01195 00614 00241 00070 00011 00001 00000 j 16 0 04401 01853 00743 0(j11i1 00100 00033 00010 00003 000017 070)5 05711 04156 01841 01654 00802 00100 00075 0G009 00000
8 08666 oml 06470 04995 03457 01060 00991 00341 00065 00003 I 08108 05147 02839 01-101 00))5 00261 00098 OOOll 00010 9 09519 09071 OJUI4 07117 05794 Q4157 Ql527 01180 00342 OOOll 2 09571 07892 05614 0)5IbJ 01gt11 00994 00451 00183 00066
10 09888 09731 09411 08868 07975 06674 04983 OlO8O 01339 00245 1 09930 09116 07899 0SJ31 04050 02459 01139 00651 00281 I 09983 09951 09874 09704 09363 08733 07664- 06017 03787 01354 4 09991 Q9830 09209 07901 1Ibull302 04-499 02891 01666 00853 12 09999 09996 09987 09961 09903 09761 0H50 U791 07458 04867 5 09 09967 09765 09 liB 08103 06598 04900 03288 01976 13 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 6 10000 09995 09944 O97i3 OZi204 08247 06B81 05272 03660
14 0 00001 00000 00000 00000 00000 00000 00000 00000 00000 00000 7 10000 09999 09989 0993u 09729 09256 011406 07161 05629 I 00009 0bull0001 00001 00000 00000 00000 00000 00000 00000 00000 10000 10000 09998 O99fiS O992S 09743 09329 08577 07441bull9 10000 10000 10000 09999 099amp1 09929 09771 09417 08759 I G0287 00114 00039 00011 00002 00000 00000 00000 00000 00000 2 00065 00011 00006 00001 00000 00000 00000 00000 00000 00000
10 10000 10000 10000 10000 ot))7 09984 09938 09809 09514 4 00898 00426 00175 00060 00017 00001 00000 00000 00000 00000 II 10000 10000 10000 100(1) 10000 09997 09987 09951 09B51 5 01120 01189 00581 0G241 GOO83 00011 00004 00000 00000 00000 12 10000 10000 10000 10000 10000 10000 09998 09991 09965 6 03951 0l586 01501 0G753 00315 00101 00024 0000l 00000 00000 Il 10000 10000 10000 10000 10000 10000 10000 09999 09994 7 06047 04519 OlO75 01836 00933 00183 00116 0G022 00002 00000 14 10000 10000 10000 10000 10000 10000 10000 10000 09999 8 07880 06627 osl4l 03595 Q2195 01117 00439 00115 00015 00000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000 9 09102 OJU18 ontrI O5m 04158 01585 01298 00467 00092 00004 16 10000 10000 10000 10000 10000 10000 10000 10000 10000
10 09713 09168 oa757 07795 06448 04787 03018 01465 00441 QOO4l
II 09935 09830 G9602 09161 U391 07189 05519 03521 01584 00101 12 09991 G9971 09919 09795 G9525 08990 08021 06433 04154 01530 11 09999 09998 09991 09976 09932 09821 09560 08971 07711 05113 14 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
~~
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
00005 00037
0000 I OQoII
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
ooqqo 00000
00000 00000
00000 00000 1
07922 09497
04818 07618
02515 05198
(UI1E 030
OOSOI 01637
00193 oon-4
00067 00327
00011 00123
00006 00041
3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
16 17
10000 10000
10000 10000
10000 10000
10000 10000
LOOOCi 10000
10000
10000 10000 10000
10000 10000
10000 10000
1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
shybull o w
liii~iii~isectsectiiiiiiimi~i~i~i~iiiiiiii
iii~iii~~iisectiiiiiiii~i~~i~~iiiiiiiii
III
bull I p
o
o
p
II
~ 1ft
o
o W In
m Z
5lt C
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
518 APPENDIX APPENDIX 519
TABLE Al (Contln~ed) TABLE A] (Continued)
I I 050 155 060 165 010 175 080 085 OtO U5 II I bull 005 010 015 Itlo Otsect 030 035 040 045
15 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 17 II 04181 01668 00631 oons oOOIS 00023 00007 00002 00000 I 1
00005 00037
0000 I OQoII
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
ooqqo 00000
00000 00000
00000 00000 1
07922 09497
04818 07618
02515 05198
(UI1E 030
OOSOI 01637
00193 oon-4
00067 00327
00011 00123
00006 00041
3 00176 00063 00019 00005 00001 00000 00000 00000 00000 00000 3 09912 09174 07556 05489 O3S30 01019 01028 0046-4 00184 4 00592 00255 00093 00028 00007 00001 00000 00000 00000 00000 4 09988 09779 09013 O7SIlt 05)3gt 03887 02348 01260 00596 5 01509 00769 00]]8 00124 00037 00008 00001 00000 00000 00000 5 09999 09953 09681 USHl Ul$] 05968 04197 02639 01471 6 03036 01818 00950 O04U 00152 00IM2 00008 00001 00000 00000 6 10000middot 09992 09917 0961j fll919 0n51 06188 0+478 01902
7 05000 03465 02131 011 n 00500 00173 00042 00006 00000 00000 7 10000 09999 09983 09091 09)0 0B954 078n 06405 047-43 8 06 05478 03902 02452 01311 00566 00181 00036 00003 00000 8 10000 10000 09997 O997i o90n 09597 09006 08011 06626 9 08491 07392 05968 04357 02784 01484 00611 00168 Ooon 00001 9 10000 10000 10000 (j99l~ 0961 09873 0967 09081 08166
10 09408 08796 07827 06481 04845 GlllS 01642 00617 00127 00006 10 0000 10000 10000 09919 o~9)4 09968 09880 09652 09174 II 09824 09576 09095 08273 07031 05387 0lS18 olm 00556 00055 II 10000 10000 10000 10000 09999 09993 09970 09894 09699 11 09963 09893 un9 09383 08732 07639 06OlO 03958 01841 00362 12 10000 10000 10000 10000 LOOOo 09999 09994 09975 09914 13 09995 09983 09948 09858 09647 09198 08319 06814 04510 01710 13 10000 10000 10000 10000 10000 10000 09999 09995 09981 14 10000 09999 09995 09984 09953 09866 09648 09126 07941 05367 14 10000 10000 0000 10000 IlJuoO 10000 10000 09999 09997 15 10000 10000 10000 10000 10000 10000 10000 10000 lIiOOO 10000 15 10000 10000 10000 10000 10000 10000 10000 10000 10000
16 0 I
00000 00003
00000 00001
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
00000 00000
16 17
10000 10000
10000 10000
10000 10000
10000 10000
LOOOCi 10000
10000
10000 10000 10000
10000 10000
10000 10000
1 00021 00006 0000 I 00000 00000 00000 00000 00000 00000 00000 II o 039n 01501 00536 OOHm OllO 00016 00004 00001 00000 3 00106 00035 00009 00002 00000 00000 00000 00000 00000 00000 I 07735 04503 02141 00991 ilOjl$ 0012 00046 00013 00003 4 00384 00149 00IM9 00013 00003 00000 00000 00000 00000 00000 2 09419 07338 0-4797 OiI ol3~3 00600 00136 00082 00015 5 01051 00486 00191 00062 00016 00003 00000 00000 00000 00000 3 0989 09018 07201 05010 oJOS7 01646 00783 00328 00120 6 022n 01241 00583 0on9 00071 00016 00002 00000 00000 00000 4 09985 09718 08794 011 bull 1 OSIB) 03327 01886 0OH2 00411 7 04018 02559 01423 00671 0Gl57 00075 00015 00002 00000 00000 5 09998 09936 09581 ullal 0717$ 053+4 03550 02088 01077 8 05982 04371 02839 01594 007 00271 00070 00011 00001 00000 6 10000 09988 09881 09~1amp7 01110 07117 05-491 03743 02258 9 un8 06340 04728 03119 01753 00796 00267 00056 00005 00000 7 10000 09998 09973 090]1 iJ~31 08S93 0n83 056]4 03915
10 08949 08024 06712 05100 03402 01897 00817 00235 00033 00001 8 10000 10000 09995 0995] 09DO) 090-4 08609 07368 05778 II 09616 09147 08334 07108 05501 03698 02018 00791 00170 00009 9 10000 10000 09999 09991 0914 09790 09403 08653 07473 11 09894 09719 09349 08661 07141 05950 04019 02101 00684 00070 10 10000 10000 10000 09990 tl99Btl 09939 G9788 09424 oa710 13 09979 09934 09817 09549 09006 08029 06481 04386 021011 0Gf29 II 10000 10000 10000 10000 09990 09986 09938 09797 09463 14 09997 09990 09967 09902 09739 09365 08593 07161 04153 01892 12 10000 10000 10000 10000 10000 09997 09986 09942 09817 15 10000 09999 09997 09990 09967 09900 G9719 09257 08147 05599 13 10000 10000 10000 10000 10000 10000 09997 09987 G9951 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 0000 14 10000 10000 10000 IJlOOO 10000 10000 10000 09998 09990
15 10000 10000 10000 10000 10000 10000 10000 10000 09999 16 10000 10000 10000 10000 10000 10000 10000 10000 10000 17 10000 10000 10000 0000 10000 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000
shybull o w
liii~iii~isectsectiiiiiiimi~i~i~i~iiiiiiii
iii~iii~~iisectiiiiiiii~i~~i~~iiiiiiiii
III
bull I p
o
o
p
II
~ 1ft
o
o W In
m Z
5lt C
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
shybull o w
liii~iii~isectsectiiiiiiimi~i~i~i~iiiiiiii
iii~iii~~iisectiiiiiiii~i~~i~~iiiiiiiii
III
bull I p
o
o
p
II
~ 1ft
o
o W In
m Z
5lt C
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
TABLEAl (Continued)
II I II 050 055 060 065 070 075 080 oa5 090 091
19 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 1 00004 00001 00000 00000 00000 00000 00000 00000 00000 00000 J 00011 00005 00001 00000 00000 00000 00000 00000 00000 00000 4 00096 00028 00006 00001 00000 00000 00000 00000 00000 00000 5 0Ol18 00109 OOOll 00001 00001 00000 00000 00000 00000 00000 6 00835 00341 00116 00031 00006 00001 00000 00000 00000 00000 7 01796 0j)871 00351 00114 00028 00005 00000 00000 00000 00000 8 03238 01841 00885 00347 OOIOS 00023 00003 amp0000 00000 00000 9 05000 Oll90 01861 00875 00316 00089 00016 00001 00000 00000
10 06761 05060 03315 01855 00839 00217 00067 Ooooa 00000 00000 II 08204 06831 05122 on 01820 00175 0OD3 00041 00003 00000 11 09165 08273 06919 05188 03345 01749 00676 00163 00017 00000 IJ 09682 09223 08371 07031 05161 03311 01631 OOS37 lt00086 00002 14 09904 09720 09304 08500 07178 05346 03167 01 00352 00020 15 09978 09923 09770 09 08668 07369 05449 03159 01150 00132 16 09996 09985 09945 09830 09538 08887 07631 05587 02946 0066S 17 10000 09998 09992 09 09896 096 09171middot 08015 05797 02453 18 10000 10000 09999 09997 09989 09958 09856 09544 08649 06116 19 10000 tOOOO 10000 10000 10000 10000 10000 10000 10000 10000
10 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 I 00000 00000 00000 00000 00000 00000 00000 00000middot 00000 00000 1 00002 00000 00000 00000 00000 00000 00000 00000 00000 00000 3 00013
~ 00003 (10015 ~0064
00000 00003 00016
00000 00000 00003
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
00000 00000 00000
r 0~0577 00214 00065 00015 00003 00000 00000 00000 00000 00000 7 01316 oOS8O 00210 00060 00013 00002 00000 00000 00000 00000 8 02517 01308 00565 00196 00051 00009 00001 00000 00000 00000 9 04119 01493 01275 00532 00171 00039 00006 00000 00000 00000
10 05881 04086 Gl 7 01218 00480 001]9 00026 0D002 00000 00000 II 07483 05857 0 02376 01 III 00409 00100 00013 00001 00000 11 08684 07480 05841 03990 0n77 01018 00321 00059 00004 00000 13 09413 08701 07500 05834 039lO 01142 00867 00219 00024 00000 14 09793 09447 087 075-46 05836 0l828 01958 00673 001ll 00003 II 09941 09811 09490 08818 01625 05851 03704 01702 00431 00026 16 09987 09951 09840 09556 08929 07741 G5886 03523 01330 00159 11 09998 09991 09964 09879 09645 09087 07939 05951 03231 00755 18 10000 09999 09995 09979 09924 09757 09308 082 0608l 01641 19 10000 10000 10000 09998 09991 09968 09885 09612 08784 06415 10 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
bullYhu tha binomial dlaoibudon wid --n n lind II The entr1q tha wluu 0I1(Y Ii n- E~ 0) 1 (I - II for II IUIJ InJ tram OJIS m 095 far n laraer than lO tha nil qllllldle 01 a IIInomIll rlndam VIII1aIH mar be IpIIIOlIImated IIIIinI t - IIJ + z VIIJ( I - p) whera Z II tha nh qIIIIIdIe 0I11Ca11d1rd RCII1IIIII rIndam wrtabl abullned tram Table AI
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
20
TABLE A7 Quantiles of the Mann-Whitney Test Statistic
1 4 5 6 10 II 12 13 14 IS 16 17 18 19n 81=2
0001 3 3 J 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0005 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4
2 001 ootS US 0
3 3 3 3
3 3 1lt4
3 3 3 4
3 3 4 middot5
3 3 4 S
3 3 lt4 S
3 4 S 6
3 4 S 6
3 4 5 7
3 5 5 7
3 5 6 8
4 5 6 8
4 5 7 8
4 5 7 9
ltI 5 7 9
4 6 7
10
4 6 8
10
5 6 8
11
5 6 8
II
bull 801 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 0005 6 6 6 6 6 6 6 7 7 7 8 8 8 9 9 9 9 10 10 001 6 6 6 6 6 7 7 8 8 8 9 9 9 10 10 II II II 12 ootS 6 6 6 7 8 8 9 9 10 10 II II 12 12 13 13 14 14 15 005 6 7 7 8 9 9 10 II II 12 12 13 14 14 15 16 16 17 18 818 7 8 8 9 10 II 12 12 13 14 15 16 17 17 18 19 20 21 22
0801 0805
10 10
0 10
10 10
10 10
10 II
10 II
10 12
10 12
II 13
II 13
II 4
12 14
12 15
12 16
13 16
13 17
14 17
14 18
14 19
001 0025
10 10
10 10
10 II
II 12
12 13
12 14
13 IS
14 15
14 16
IS 17
16 18
16 19
17 20
18 21
18 II
19 II
20 23
20 24
21 lS
005 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 lS 26 27 28 29 010 II 12 14 15 16 17 18 20 21 22 23 2lt1 26 27 28 29 31 32 3l
0001 IS 15 IS 15 IS 15 16 17 17 18 18 19 19 20 21 21 22 23 23
5
ooos 001 0025
15 IS 15
15 IS 16
15 16 17
16 17 18
17 18 19
17 19 21
18 20 22
19 21 II
20 22 24
21 23 lS
22 M 27
23 lS 28
23 26 29
24 27 30
lS 28 31
26 29 33
27 30 34
28 31 35
29 32 36
US 16 17 18 20 21 22 24 lS 27 28 29 31 32 34 35 36 l8 39 41 010 17 18 20 21 II 24 26 28 29 31 II 34 36 38 39 41 43 46
8801 21 21 21 21 21 21 II 24 lS 26 26 27 28 29 30 31 32 II 34 0805 21 21 22 II M lS 26 27 28 29 31 32 33 34 35 37 38 39 40
6 001 0025 us
21 21 22
21 II 24
II M lS
M lS 27
lS 27 29
26 28 30
28 30 32
29 32 34
30 33 36
31 35 38
33 36 39
34 38 41
35 19 43
37 41 4S
38 43 47
40
41 46 50
42 47 52
49 54
011 23 lS 27 29 31 33 35 37 39 41 43 45 47 51 53 56 58 60
0801 ooos
28 28
28 28
28 29
28 30
29 32
30 n
31 3S
32 36
34 38
35 39
36 41
37 42
38
39 45middotmiddot
40 47
42 43 50
51
45 53
7 001 0025
28 28
29 30
30 32
32 34
33 35
35 37
36 39
38 lt41 43
41 45
43 47
45 49
46 51
53
50 55
52 S7
53 59
55 61
57 63
us 010
De1 0805
29 JO
36 36
31 33 36 36
33 35 36 38
35 37
37 19
37
38 41
lt42 39 43
42 45
41
47
42 46
46 so 43
52 45 so
so 55 46 52
53 57
54
55 60 49 55
57 OJ 51 57
59 65 52 59
62 67
54 61
64 70 55 63
66 72 57 65
68 75
58 67
bull 001 U25
SLlO
36 37
3839
37 19
lt42
39 41 42
41 43 45 47
43 lt45 lt47 50
47 so 53
46 SO 52
~
ltII 52 55 59
so 54 57 61
S2 56 60 64
54 59 37
56 61 65 70
59 63 68 73
61 66 70 76
63 68 73 79
65 71 76 82
67 73 78 85
69 75 BI BB
71 78 84 91
TABLE A7 (Continued)
n m e 2 5 6 7 bull 9 10 II 12 13 I 15 16 17 18 19 20
9
10
II
II
Il
14
15
1001 0005 061 0025 005 010 0001 080S OCII D025 005 010
0801 0005 DOI 0025 005 010 0001 OOOS 001 0025 005 016
0001 000 001 COlS 005 010
0001 0005 001 0825 005 010 0001 0005 001 0025 005 010 0001 0005 001 0025 005 010
45 45 45 46 47 55 55 55 56 57 59
66 66 66 67 6B 70
78 78 78 80 81 83 91 9l 91 93 94 96
105 105 106 107 109 110
120 120 12l III 124 126
136 136 137 138 140 142
45 46 47 48 50
5 55 56 57 59 60 62
66 67 68 70 72 74 78 80 81 83 84 S7 Sf 93 94 98
101
lOS 107 lOB III 1t3 116
120 III 124 126 128 III 136 139 1 143 45 148
45 47 so 52 55 56 58 59 61 63 66
67 69 71 73 75 78
79 82 84 86 88 91
93 95 97
100 102 105
107 110 112 115 117 121
III 126 128 131 133 137
139 142 I 151 154
474 51 53 55 68
57 60 OJ 64 67 69
69 72 74 76 79 82
81 65 87 90 92 96 95 99
101 104 107 110
109 113 116 119 122 126
125 129 132 135 139 143
142 46 149 152 156 160
48 51 53 56 68 61 59 62 64 67 70 73
71 74 76 80 83 86
83 86 90 93 96
100
97 102 104 108 III ItS
112 117 1t9 III 127 131
128 133
136 140 148
145 ISO 53 168 162 166
49 53 55 58 61 64
61 65 67 70 73 77 73 77 79 83 86 90
86 91 93 97
100 105
100 105 108 112 116 120
115 121 123 128 132 137
133 137 140 145 149 154
148 ISS 158 163 167 173
51 55 57 61 64 68
62 67 69 73 76 80 75 80 8l 86 90 94
88 94 96
101 105 lOS
103 lOS 112 116 120 125
118 124 128 132 137 142
135 141 145 150 154 160
152 159 163 168 173 179
53 57 60 63 67 71 64 69 72 76 80 84
77 63 65 90 94 98
91 97
100 105 l09 II~
106 112 115 120 IlS 130 121 128 132 137 142 147
138 145 14 ISS 160 166 156 164 168 174 179 185
54 59 62 66 70 74
66 72 75 79 83 BB 79 85 89 93 98
103
3 100 103 lOB III liB 109 116 119 125 129 135
125 132 136 142 147 153
112 150 154 160 165 172
160 168 173 179 185 191
56 62 64 69 73 77 6B 74 78 8l 87 91 8l 88 92 97
101 107
96 103 107 112 II 123
112 119 123 129 134 140
128 [36 140 146 152 168
145 154 158 165 171 178 164 173 178 184 191 198
58 64 67 72 76 81 70 77 80 85 90 95
84 91 95
100 105 III 96
106 110 116 111 128
I IS 123 127 133 139 145
131 140 I 151 157 164
149 158 163 170 176 184
168 178 183 190 197 204
60 66 69 74 79 84 73 80 83 89 93 99 87 94 98
104 109 115 102 110 114 120 126 132
liS 126 III 137 1~3 ISO
135I 149 156 162 169
153 163 168 175 18l 189
172 182 188 1 202 211
61 6B 72 77 8l 87 75 82 86 91 97
103 89 97
101 107 113 119 104 113 117 124 130 137
121 130 135 142 146 155
13B 148 153 161 167 175 157 167 172 180 187 195
176 187 193 201 208 217
63 70 74 80 85 91 77 85 89 95
100 107
100 104 III 117 124 106 116 121 128 134 142 124 134 139 146 153 160
142 152 157 165 172 180
161 172 177 185 193 201
180 192 198 207 214 223
65 73 77 83 88 94 79 87 92 98
104 110 94
103 lOB 114 121 128 110 120 125 132 Il9 146 127 137 143 151 157 166
145 156 162 170 177 186
164 176 182 191 198 207
185 197 203 212 220 230
67 75 79 85 91 98
81 90 94
101 1D7 114
96 106 III 118 IH 132 f 3 123 128 136 J43 ISl
130 141 147 155 162 171
149 160 166 175 IB3 191
168 IBI 187 19 204 213
189 202 20B 218 226 236
69 77 8l 88 94
101
83 93 97
104 III 118
99 109 114 122 126 136
J 16 126 131 110 147 1S6 1)4 15 151 159 167 176
152 164 171 180 IBB 197
In 185 191 201 209 219
193 207 113 223 232 243
71 79 84 91 97
104
85 95
100 108 114 122
101 112 117 125 131 140
118 130 135 144 151 160
137 1lt9 155 164 172 IBI
156 169 175 164 193 203
176 190 196 206 215 225 197 211 219 229 238 249
72 82 86 94
100 lOB 88 98
103 III 118 126
104 115 120 129 136 145
121 133 139 148 156 r65
140 152 159 168 176 186
160 173 179 189 198 208
180 194 201 211 221 231
202 216 224 235 2 256
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
t
TABLEA7 (Continued)
bull -2 I 4 J 7 bull 9 10 II 12 II 14 15 17 18 21 1middot 153 IH 156 159 163 167 171 115 179 183 188 191 197 2111 206 III 115 l2D 11~
17 UIIS U2I
153 IH 156
156 158 160
160 162 165
164 167 171
169 172 176
173 171 182
178 182 188
III 187 193
188 192 199
193 198 2D5
198 213 211
203 2D9 217
2Il8 21lt1 m
21~ 220 129
219 llS 235
11~ 231 211
119 236 2lt17
235 2lt12 253
210 217 159
us 157 163 169 171 180 187 193 199 2D5 211 218 22lt1 231 237 2lt13 250 256 163 269 011 160 166 172 179 185 192 199 206 III 219 ll6 233 239 1lt16 153 260 267 2H 281 1 171 172 115 178 182 IN 190 195 199 lOltI 2D9 111 218 m 1lI 233 238 213 l~8 UIIS 171 17~ 178 III 188 193 198 213 2M 214 219 llS 230 ll6 242 2lt17 253 259 2M
10 11 DD2J
172 174
176 179
181 184
IN 190
191 196
196 2D2
2D2 2Il8
2Il8 214
213 220
219 227
225 233
231 239
237 2lt16
242 252
2lt18 25B
25~ 265
260 271
166 278
272 284
us 176 181 188 194 200 2117 213 220 117 233 2lt10 247 159 260 JJ7 274 lBl 188 295 0bullbull1 178
190 185 191
192 194
199 198
206 2D2
213 206
220 211
227 216
134 220
2lt11 225
249 231
256
ll6 JJ3
241 270 2lt16
278 251
285 257
292 JJ2
300 168
307 272
DM5 19 194 198 213 2Il8 213 219 l34 230 236 242 2lt18 159 260 JJS 272 278 284 290 It
11 U2I
192 193
195 198
200 lOltI
206 liD
211 216
217 m
m 129
129 ll6
m 243
2lt11 249
2lt17 256
159 163
260 269
166 276
273 lB3
279 290
2B5 297
292 301
298 310
15 195 2111 2Il8 211 111 1lI m 2~2 249 256 263 271 278 285 292 300 307 ll4 321
20
81 1D2J
198 210 211 212 211
2115 211 214 216 219
212 214 219 221 225
219 21B 22lt1 l27 231
l27 m 219 231 238
134 l27 235 219 2lt15
242 132 241 2lt15 251
249 217 217 251 259
257 243 253 258 166
2M 248 259 2M 273
272 253 l6S 27 280
280
159 271 278 2B7
188 l6S 278 284 2M
295 270 284 291 301
303 276 290 298 30t
311 281 297 301 316
ll9 2B7 303 311 323
326 293 310 31B llO
334 299 316 325 338
us 215 m 229 ll6 243 l50 258 l6S 273 280 188 295 303 311 318 326 334 341 349 111 218 ll6 l3l 241 2lt1 257 l6S 273 281 189 297 lOS 313 l21 330 l38 346 lH l62
For II or m duIn 20 the fICh quamfIe w of the IIaM-WhIIMy rut stadsdc may be ~ by
(11+ 1)2 + zV_(N + 1)12
where z1s the fICh quuuIe of mndatd normal random wriabIe abaIIned from Table AI and where 11 m + IL
The __ In dIiIllIbIe quandlas w of the Mann-WhIIMylllt JIaIIsdc r by Equation 511 for selected values of Note that p(r lt wl S p Upper quanshy dies may be found from the equaDon o
w IIII + m +1) - w-
Critical COIIeSpOIICI to values of r less duIn (or duIn) but not equal to the appropriate quantile
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ bullbull ww-ww-ww-w-ww-ww-w-ww-w-w
( ~
f
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
542 APPENDIX
TABLEAIO Quantiles of Spearmans P
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000 6 7
06000 05357
07714 06786
08286 07500
08857 08571
09429 08929 09643
8 05000 06190 07143 08095 08571 09286 9 0A667 05833 06833 07667 08167 D9000
10 04424 05515 06364 07333 07818 08667 II 0A182 05273 06091 07000 07455 08364 12 03986 04965 05804 06713 07203 0amp112 13 03791 04780 05549 06429 06978 07857
14 03626 04593 05341 D6220 06747 07670 15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 07265 17 03260 04118 0A853 05637 06152 07083 18 03148 03994 04696 05480 05975 06904 19 03070 03895 04579 05333 05825 06737 20 02971 03789 04451 05203 056amp4 06586
21 02909 03688 04351 05078 05545 06455 12 02829 03597 04241 04963 05426 06318 13 02767 03518 04150 04852 05306 06186 14 02704 03435 04061 04748 05200 06070 25 02646 03362 03971 04654 05100 05962
26 02588 03299 03894 04564 05002 05856 27 02S40 03236 03822 04481 04915 05757 28 02490 03175 03749 04101 04828 05660 29 02443 03113 03685 04320 04744 05567 30 02400 03059 03620 04251 04665 05479
For n greater than 30 dI approximate quantlles of p may be obcalned from
wmiddot V Zjon-I
where Zjo Is the fJdI quantile of a standard normal random variable obmlned from Table AI Souaa Adapted from Glasser and Winter (1961) with comedo with permlulon from die 8omerrIkQ
Tru_ bullThe entries In dli table are selected quantllu w of die Spearman rank correla1lon coaffIcIent p when IIMd as a teSt statistic The lower quantllbullbull may be obalned from die equadon
W == -WI_
Th crtdal region correspOnds to values of p maller dI (or areater than) but not Including dI approshypriate quantile Note dlat the median of p Is O
4 4 (06667) 5 6 (06000) 6 7 (04667) 7 9 (0A286) 8 10 (03571) 9 12 (03333)
10 15 (03333) II 17 (03091) 12 18 (02727) 13 11 (02821) 14 23 (02527) 15 27 (02571) 16 28 (02333) 17 32 (02353) 18 35 (02288) 19 37 (02164) 10 40 (02105) 21 42 (02000) 12 45 (01948) 23 49 (01937) 24 52 (01884) 15 56 (01867) 26 59 (01815) 27 61 (01738) 28 66 (01746) 29 68 (01675) 30 73 (01678) 31 75 (01613) 32 80 (01613) 13 84 (01591) 34 87 (01551) 35 91 (01529) 36 94 (01492) 37 98 (01471)
APPENDIX smiddot
TABLE AI I Quantiles of the Kendall test statistic T = N - N bullbull Quantlles of Kendalls T are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n p = 0900 0950 0975 0990 0995
4 (06667) 6 (10000) 6 (10000) 6 (10000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 9 (06000) II (07333) II (07333) 13 (08667)
II (05238) 13 (06190) 15 (07143) 17 (08095) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 16 (04444) 18 (05000) 11 (06111) 24 (06667) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 24 (03636) 28 (04242) 34 (05152) 36 (05455) 26 (03333) 32 (0403) 38 (04872) 42 (05285) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 40 (02941) 48 (03529) 56 (004118) 62 (04559) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 50 (02632) 60 (03158) 70 (03684) 78 (04105) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 70 (02333) 84 (02800) 98 (03267) I08 (03600) 75 (02308) 89 (02738) 105 (03231) f15 (03538) 79 (02251) 93 (02650) II I (03162) 123 (03504) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 93 (02138) 109 (02506) 129 (02966) 143 (03287) 97 (02086) 115 (02473) I3S (02903) 149 (03204)
102 (02056) 120 (02419) 142 (02863) 158 (03185) 106 (02008) 126 (02386) 150 (02841) 164 (03106) III (01979) 131 (02335) 155 (02763) 173 (03084) 115 (01933) 137 (02303) 163 (02739) 179 (03008) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 126 (01892) 150 (02252) 176 (02643) 198 (02943)
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
II
i AlPIHDDC
rABLEAIi (Continued)
OtoO 0910 097S 0990 O99S
I APPENDIX 55
TABLE All Quantile of the Wilcoxon iZi2gtIilI Wmlili Ylilst Statlsdc - - -- ~
11(11 + I) W wu Wuu Wus tiiJJu IjU1I Wuo WOM W 1
II 19 10 II 12 11 14 II 16 17 II 19 10 II i2 11 jot II 16 17 II i iO
103 (01-465) 107 (01+14) 110 (01372) 11-4 (01390) 119 (01381) 113 (01361) 128 (01353) III (01333) 135 (01304) 141 (D131H) I (01277) 150 (01276) IS] (012-49) 159 (012-47) 162 (01111) 168 (01219) 173 (01109) 177 (01192) 182 (01181) 186 (01165) Itl (01155) 197 (01151) 101 (01141)
111 (01863) m (01849) 1-41 (01811) 1-46 (01780) 151 (01754) 157 (01739) 161 (01711) 168 (01697) 173 (01671) 179 (01656) 186 (01649) 190 (01616) 197 (01608) 103 (01592) 208 (01569) 21-4 (01553) 121 (015 ) 117 (Otn9) 232(01506) 1laquo) (OISIH) 25 (01-481) lSI (01-467) 158 (0158)
ISS (CUlO5) 161 (01171) 168(0115-4) 17-4 (01111) 181 (01101) 187 (01071) 194 (0l0S1) 200 (01010) 207 (01000) 111 (01970) 110 (01950) 118 (01919) 131 (01901) 2-41 (01890) 2-ta (01870) 156 (01858) 263 (01838) 269 (01811) 276 (01792) 18-4 (om9) 291 (01760) 199 (017lt18) 306 (01729)
183 (G2603) 191 (0lS78) 198 (0l538) 106 (01511) 113 (01-47) 111 ~1+r1)118 2 10) 236 (02383)
l-t5 ~01367)lSlOll4O) 260 (01305) 168 (01179) 2n (01161) 185(OlllS) 294 (01117) 101 (01192) 111 (01173) 319 (021-tB) 328 (01110) 336 (01105) l-tS (01087) 355 (01075) 364 (01056)
103 (01888) 111 (018-48) 110 (01811) 118 (01780) 135 (01729) 1-4S (01713) lSl (01664) 262 (01646) 171 (02618) 279 (0l581) 288 (02551) 296 (01517) lOS (01-490) 315 (01471) ll (013) 134 (01-424 341 (01397) 351 (02377) 161 (01351) 372 (01331) 381 (01305) 391 (01185) -tOl (01171)
or I tv chilli 60 ~ quIIIdlu of T II1II) be obDltMd from
bull -11- 111la + 51w Z 18
haN ~ from rha IIW1IIInIIIOITIIIII dIIIrIbudon - by Table AI ~ -cJ1u 01 1II1II) b from
~+5lwmiddotZJ~_I)
rtdcaI coftupond CD wIuu of T dIIIII (ar dIIIII) but _ JncIudInc rha approprtat Not thac ch IMdIIn of T II 0 QIIIIUlIu for I _ obtaIMd by dlvldIIII rha cpndIu of T by (n - 1)12
OUICI Adapud from Tab I But (1914) wid )1IIIIIIon Irom rha author
ft-S 6 7 I 9
10 II 12 11 14 II 16 17 18 9 10 21 21 U 24 21 16 17 11 19 10 11 II 3l J4 31 16 J1 18 19 40 4 42
0 0 0 0 I 1 -4 6 8
0 11 16 10 24 28 31 18 49 55 62 69 76 84 92
101 110 119 129 139
H 160 172 184 196 lOB
221 235 2-ta
0 0 0 I 1 -4 6 8
10 13
16 10 14 18 3l 38 50 56 63 70 77 85 94
101 III 121 III 41 152 163 175 187 199 212 m 239 lS3 267
0 0 1 1 -4 6 9
II 18 11 16 30 35 -41 47 53 59 67 74 81 90 99
lOB 117 127 138 l-ta 160 171 183 196 109 m 236 lSD 165 280 295
0 I 3 -4 6 )
II 14 III 21 26 31 36 42 48 54 61 68 74
84 92
101 III 120 131 141 152 161 176 188 201 214 228 142 257 272 287 10] 320
j
-
shy)
II IS 12
~
t
r
-D -i
~G
ij
~f(
iii L~
is lOt
u-) I~
125
I~i
150 110
1iJ1 195 Oo 212 B6 2S1 266 101 1)11
31-l 331 349
3
-J ~
)
1- IS jSl
~j
2bi jj
Jl)
~~
~i
Z
)-
()shy
middot1 IOU iu Igtu UI H ~ l- -I- IcC Ilu 191 10~
219
m 2JiJ 1oj
17
29$
311
J2J
Jmiddotfi 3amp5 3lJ
3 5 B
II 14 18 n 27 32 38 4-4 51 58 65 73 82 91
100 110 no 131 143 ISS 167 180 193 207 211 236 151 266 283 299 316 334 352 371 390 409
-4 6 9
12 16 10 25 30 36 41 -t8 55 63 71 80 89 98
108 119 130 1 1 153 165 178 191 206 220 215 250 266 182 299 317 335 353 372 191 411 431
5 75
105 H 18 215 275 33 39 55 525 60
68 765 855 95
IDS 1155 1265 138 150 1625 1755 189 201 2175 2325 2-t8 264 2805 2975 31S III 351S 3705 390 410 4305 -tS15
10 IS 21 28 36 45 55 66 78 91
105 120 116 153 171 190 210 231 lS3 176 100 325 3S1 378
35 --465
528
561 595 630 666 703 741 780 820 861 903
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
54 APPENDIX
TABLE All (Contlnuad)
n(n+ I) WUII 111- 111 1111bullbull Wou Wue WIIM 111 2
43 26l 28l lit 337 ]66 lt4Ol 429 452 473 946 277 297 ]28 ]54 385 n so 473 495 990 45 292 311 1 In 01 2 ltfT1 95 5115 1035 108 129 161 390 23 -463 492 511 5ofO5 1081 7 32 3-46 379 of08 2 51 5-40 564 1128 3ofO 36l 397 28 -463 505 536 563 588 1116 9 351 381 416 f83 521 559 587 6125 IllS 50 31 398 35 61 5Oof 550 583 611 6315 1215
For n Iarpr diu so the fgtth qlllll cI the WIlcoxon lIped _ IGdIIII may be apprcPdmaIM by ~ - [nn + 1)14] + Ifvn(n + 1)(2n + l)l wheN If II the fgtth qqand cia ~d IIOrmailllldam wrIIbItI obaInad from Table AI SouIIcL Adapud from Hamr and o-n (1970) with ~ from tha AmeriCIII ~ SodMy The mr1 In thl1IIIIIe q cI the WIlcoxon Ifaned IIIIkI_1GdIIII r JIYIIII by Equadon 573 for HIKtad wishy_ of P so 050 Quancll w far p gt 050 may be campuud from the IlljUldQII
w - n(n + )1- 11bull
where n(n + 1)111 pen In tha rIampht hand column In tha tabl N_ dItIl I(r lt w) so and I(r gt w ) so I - II Ho II IIIIL Crllical corrapcad co 1111I11III rI r dian (or amprater diu) 1M not IncIucIInI the 9Proprllle quantlla
i
n =I 0900 0950 2 0684 0776 1 0565 0616 093 0565 5 0
MIO 0509 068
7 0181 036 a 0358 010 9 0119 0387
10 Onl 0369 II ol08 0152 12 0296 0ll8 Il 0285 0325 I 0275 031 156 Q266
0258 0304 0295
17 0250 02116
bullbull920
02 0237 0ll2
0219 0271 0265
APPENDIX 547
~ TABLEAIJ Quantll of the Kolmogoiov Y~lC S~ti5tkmiddot
OnSlded Tet P - 090
Two-Sided Test P =080
195
090
0975
195
099
090
099$
Ui1l
P 090
== 000
095
090
0975
095
099
098
0995
099
0975 0990 0995 ~iI = u 0226 08-42 0900 0929 lJ 0221 D708 0785 0029 0216 062 0689 0734 ~~ 0212 0563 0627 0669 ~S 0208 0519 0577 0617 ~ 0204 083 0538 0571 middott 0200 0 5 0507 0H1 w 0197 010 0480 0511 LJ 0193 MOt 0457 010) 311 0190 0191 0437 046amp 21 0187 0l75 0419 0449 r2 0184 0361 0404 0-132 J~ 0182 039 0390 041amp j~ 0179 0138 0377 0404 j~ 0177 0321 0366 0391 ~i 0174 0318 0355 0381 1 0172 0309 0346 0371 iu 0170 0301 0337 0361 J() 0168 02 0329 0352 ~~U 0165
Approltimatloh fOlngt 40
107 Va
0259 0287 0321 0344 0253 0281 03 0337 0247 0275 0307 0330 0242 0269 0101 0323 0138 0264 0295 0317 0233 0259 0290 0311 0229 0254 Ol8of 0305 0225 0250 0279 0300 0221 0246 0275 0295 0218 0242 0270 0290 021 0218 0266 0285 0211 023of 0262 028 0208 0231 0258 0177 0205 0227 025 0273 0202 022 0251 0269 0199 0221 027 0265 0196 028 0244 0262 01 0215 021 0258 0191 0213 0238 0255 0189 0210 0235 0252 122 136 152 163 Va Va Va Va
5auIIcE Adapud from Table I of HW (1956) Used with permission 01 tho American Stadatlcal AlISoclatlon
Themr1a In thII table lite uliacted quantllti w of the Kolmogorov ten tatlmcs r p and r- defined by Equation 611 for two-IIded _ and by EIjIIIIIoRI6Il and 61l far onemiddotided te Rejeelt H at the Ivel If r xcead$ tho I - quantile p In thII tabla TMH quanlhl ant _ far II S ~O in the cwomiddottailed ast The oth quantll1 are approximation that aqual cothe -1 quantIIa In mOlt casu A better approximation for Il gt 40 results If (n + VnIiO)lIl I Jlsed InfAd cI Yn In the danomlnator
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
556 APPENDIX APPENDIX
TABLE A20 Quantiles of the Smlrnov Test Statistic for Two Samples of Different Size nand m
TABLE AI9 Quantile of the Smlrnov Test Statistic for Two Samples of Equal Size n One-Sided Test p =090 095 0975 099 0995 Two-Sided Test p =080 090 095 099 099
One-Sided Test One-SIded Test II = 090 G95 0975 099 0995 II = 090 095 0975 099 0995 11 = I Il = 9 17118
10 9110Two-SIded Test Two-Sided Test II =080 090 095 098 099 II =080 090 095 098 099 11 =2 Il = 3 516
4 34 10122 1011l 5 45 45 1023 10123 6 516 56 10114 1I1l4 7 517 617
8 34 718 78lOllS 1I1l5 9 719 819 81910126 11116
10 7110 45 9110 11 = 3 H = 4 34 3411117 11117
5 213 45 41511118 12128 6 213 213 5611119 12129 7 213 57 617 61711130 12130 8 58 34 34 71811131 12131 9 213 213 719 819 81912132 12132 10 35 7110 415 9110 9110
12 7112 213 34 56 111212133 1333 11 = 4 H = 5 315 314 45 415 12134 133 6 7112 213 34 56 5612135 1335 7 17118 517 34 617 617 12136 1336 8 518 518 3 78 718 1337 1337 8199 519 213 31 719
10 11110 13110 710 45 451338 14138 12 7112 213 213 314 561339 1439 16 9116 58 1116 3 131613lt10 141lt10
11 =5 H = 6 3S 213 213 516 516215 230 7 417 2335 517 29135 617
n=3 4 5 6
it 7
8 9
10 II 12 13
14 15 16 17 18 19 20 21
213 314 35 36 417
418 419 410 5111 5112 513
5114 5115 6116 6117 6118 6119 6120 6121
213 314 35 46 417
48 59 5110 5111 512 6113
6114 6115 6116 7117 7118 7119 7120 7111
314 45 46 517
518 519 6110 6111 6112 6113
7114 7115 7116 717 8118 8119 8110 821
45 516 517
58 619 6110 7111 7112 7113
714 8115 8116 8117 918 9119 9110 9111
n =22 23
415 24 56 15 517 26
68 27 619 28 7110 29 7111 30 7112 31 8113 32
8114 33 8115 34 9116 35 9117 36 918 37 9119 38
10110 39 10111 40
Approximation for n gt 40
711l 7123 7124 7125 7126
7127 8128 829 830 8131 8132
8133 8134 8135 9136 9137 9138 9139 91lt10 Lg Vii
8122 823 8124 825 8126
8127 9128 9119 930 9131 9132
9133 10134 1035 1036 1037 10138 10139 101lt10 m Vii
8122 923 9114
gt91l5 9126
9117 10128 10129 1030 10131 10132
11133 11134 11135 11136 11137 1138 1139 121lt10 192 Vii Vii Vii 8 11110 58 27140 41S 415
9 519 35 31145 719 415Souroa Adapted from Birnbaum and HaH (1960) wid permluJon from the Institute of Mathematical StatlstIes I III 35 710 7110 45bullThe entries In this table ant selected quantlles w of the Smlmov lWCI-AIIlple _ statistic T deflned by Equation 632 and 633 for the one-taIled _ and defined by Equation 631 for the _tailed test Relect He at the level a If T exteeds the I - a quanshy 15 8IIS 35 213 IIIIS 1115
20 III 11110 3S 7110 34tile of T as slYen In this table The _ statistic Is a dIscrere random variable so the exact level of snlll_ce may be 1_ than the apparent a used In this table 11 =6 H = 7 2342 417 29142 517 56
8 III 7112 213 31 349 III 59 213 1318 79I III 1730 19130 7110 11115
12 III 7112 7112 213 3418 49 519 11118 213 13118 24 1I1l4 III 7112 SIB 213
IiJ
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p =06 075 09 095 0975 099 0995 09975 0999 09995
APPENDIX
A10 (Continued)
099 0995~ed Test p = 090 095 0975 099 099~ed Test P = 080 090 095
N1 = 8 9
10 14 28
N1 = 9 10 12
32 M= 10
12 15 18 36
N1 =15 20 40
M=15 16 18 20
N1 =20 Nz = 20
mple nadon
2756 3163 33nO 3n 3n o
1940 11114 716
1332 7115 419
1945 7118
1336 215 215 7110
2360 38
1336 11130 7110
2780
107jm+n mn
3356 519
3970 III
13128 13124
middot21140 III III 716 III III
22145 49 5112 715 9110 215 9110 716 Sll2 512 215
31180
l11jm + n mn
58 4063 43nO
417 15118 518
2340 7112 916III
26145 519 8115 III
1736 III III 9110 III
2348 1736 7115
13130 1740
136jm n
41156 5n 7110 914
17128 213
2740 58 511 9116 213
IIIUI 35 519
19136 1730 11110 III
11110 13114 19136 31160 29160
19140
152jm + n mn
34 47163 5n 517 9114 34 710 213 518
19131 3145
213 29145 11118 519
19130 35
7112 7112 519
1730 3160 41180
I63Jm + n mn
2 3
5 6 7 8 9
10 II 12 I]
14
15
16 17
18
19
20 21 22
23 24
25 26
27 28
29
30
40
60 120
0325 1000 3078
0289 0816 1886
0277 0765 1638
0271 0741 1533
0267 0727 1476
0265 0718 10440
0263 0711 1415
0262 0706 1397
0261 0703 1]8]
0260 0700 1372
0260 0697 1363 0259 0695 1356 0259 0694 1350
0258 0692 1345
0258 0691 1341
0258 0690 1377
0257 0689 1333
0257 0688 1330 0257 0688 1328
0257 0687 1325
0257 0686 1323
0256 0686 1321
0256 0685 1319
0256 0685 1318
0256 0684 1316
0256 0684 1315 0256 0684 1314
0256 0683 1313
0256 0683 1311
0256 0683 1310
0255 0681 1303
0254 0679 1296
0254 0677 1289
0253 0674 1282
6314 12706 ) I 821 63657 12732 31831 63662
2920 4303 6965 9925 14089 22327 31598 2353 3182 4541 5841 7A53 10214 12924 2132 2776 3747 4604 5598 7173 8610
2015 2571 3365 4032 4773 5S93 6869 1943 2447 3143 3707 4317 5208 5959 1895 2365 2998 3499 4029 4785 5408 IS60 2306 2896 3355 3833 4501 5041 1833 2262 2S21 3250 3690 4297 1781
1812 2228 2764 3169 3581 4144 4587 1796 2201 2718 3106 3497 4025 4437 1782 2179 2681 3055 3428 3930 431S 1771 2160 2650 3012 3372 3852 4221 1761 2145 2624 2977 3326 3787 4140
1753 2131 2602 2947 3286 3733 4073 1746 2120 2583 2921 3252 3686 4015
1740 2110 2567 2898 3222 3646 3965
1734 2101 2552 2878 3197 3610 3922
1729 2093 2539 2861 3174 3579 3883
1725 2086 2528 2845 3153 3552 3850
1721 2080 2518 2831 3135 3527 3819
1717 2074 2508 2819 3119 3505 3792
1714 2069 2500 2807 3104 3485 3767
1711 2064 2492 2797 3091 3467 3745
1708 2060 2485 2787 3078 3450 3725
1706 2056 2479 2779 3067 3435 3707
1703 2052 2473 2771 3057 3421 3690
1701 2048 2467 2763 3047 3408 3674
1699 2045 2462 2756 3038 3396 3659
1697 2042 2457 2750 3030 3385 3646
1684 2021 2423 2704 2971 3307 3551
1671 2000 2390 2660 2915 3232 3460
1658 1980 2358 2617 2860 3160 3373
1645 1960 2326 2576 2807 3090 3291
Iapted from Musey (1952) with permission from the Instkuce of Mathematical St3t1Rlcs
as In this tabl are sefectecl quantlas w of the SmImoY test statistic T for two samples defined ns 631 632 and 633 To enter the table let N be the smaller sample sbe and let IV be the raquoIe sbe Reject H at the leYel a If T ceeds w- as ampIvan In this table If II and m are nO( cOyenshyIs table use the fa sample approxlmatlon given at the end of the tabl or consult elCact tables j Jennrlch which appear In Haner and Owen (1970) for m S 100
Souoa Reprinted from Vol I of Pearson and Hartley (1976) with permission from the Iliomerrlka Truste bull The entries in this table are quantlles w of the t ditribution for various degrees of freedom Quantlles w for p lt 05 may be computed from the equatlon
W -Wt_jI
Note that w = 0 for all dearees of freedom
- bull amp tmiddotlh
