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726Appendix A Statistical Tables and Proofs
Table A.1 Binomial Probability Table727rTable A.1 (continued) Binomial Probability Sums b(x; n, p) x=0
Table A.1 Binomial Probability Sums
r b(x; n, p)
x=0
p
nr0.100.200.250.300.400.500.600.700.800.90
100.90000.80000.75000.70000.60000.50000.40000.30000.20000.1000
11.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
200.81000.64000.56250.49000.36000.25000.16000.09000.04000.0100
10.99000.96000.93750.91000.84000.75000.64000.51000.36000.1900
21.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
300.72900.51200.42190.34300.21600.12500.06400.02700.00800.0010
10.97200.89600.84380.78400.64800.50000.35200.21600.10400.0280
20.99900.99200.98440.97300.93600.87500.78400.65700.48800.2710
31.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
400.65610.40960.31640.24010.12960.06250.02560.00810.00160.0001
10.94770.81920.73830.65170.47520.31250.17920.08370.02720.0037
20.99630.97280.94920.91630.82080.68750.52480.34830.18080.0523
30.99990.99840.99610.99190.97440.93750.87040.75990.59040.3439
41.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
500.59050.32770.23730.16810.07780.03130.01020.00240.00030.0000
10.91850.73730.63280.52820.33700.18750.08700.03080.00670.0005
20.99140.94210.89650.83690.68260.50000.31740.16310.05790.0086
30.99950.99330.98440.96920.91300.81250.66300.47180.26270.0815
41.00000.99970.99900.99760.98980.96880.92220.83190.67230.4095
51.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
600.53140.26210.17800.11760.04670.01560.00410.00070.00010.0000
10.88570.65540.53390.42020.23330.10940.04100.01090.00160.0001
20.98420.90110.83060.74430.54430.34380.17920.07050.01700.0013
30.99870.98300.96240.92950.82080.65630.45570.25570.09890.0159
40.99990.99840.99540.98910.95900.89060.76670.57980.34460.1143
51.00000.99990.99980.99930.99590.98440.95330.88240.73790.4686
61.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
700.47830.20970.13350.08240.02800.00780.00160.00020.0000
10.85030.57670.44490.32940.15860.06250.01880.00380.00040.0000
20.97430.85200.75640.64710.41990.22660.09630.02880.00470.0002
30.99730.96670.92940.87400.71020.50000.28980.12600.03330.0027
40.99980.99530.98710.97120.90370.77340.58010.35290.14800.0257
51.00000.99960.99870.99620.98120.93750.84140.67060.42330.1497
61.00000.99990.99980.99840.99220.97200.91760.79030.5217
71.00001.00001.00001.00001.00001.00001.00001.0000
p
nr0.100.200.250.300.400.500.600.700.800.90
800.43050.16780.10010.05760.01680.00390.00070.00010.0000
10.81310.50330.36710.25530.10640.03520.00850.00130.0001
20.96190.79690.67850.55180.31540.14450.04980.01130.00120.0000
30.99500.94370.88620.80590.59410.36330.17370.05800.01040.0004
40.99960.98960.97270.94200.82630.63670.40590.19410.05630.0050
51.00000.99880.99580.98870.95020.85550.68460.44820.20310.0381
60.99990.99960.99870.99150.96480.89360.74470.49670.1869
71.00001.00000.99990.99930.99610.98320.94240.83220.5695
81.00001.00001.00001.00001.00001.00001.0000
900.38740.13420.07510.04040.01010.00200.00030.0000
10.77480.43620.30030.19600.07050.01950.00380.00040.0000
20.94700.73820.60070.46280.23180.08980.02500.00430.00030.0000
30.99170.91440.83430.72970.48260.25390.09940.02530.00310.0001
40.99910.98040.95110.90120.73340.50000.26660.09880.01960.0009
50.99990.99690.99000.97470.90060.74610.51740.27030.08560.0083
61.00000.99970.99870.99570.97500.91020.76820.53720.26180.0530
71.00000.99990.99960.99620.98050.92950.80400.56380.2252
81.00001.00000.99970.99800.98990.95960.86580.6126
91.00001.00001.00001.00001.00001.0000
1000.34870.10740.05630.02820.00600.00100.00010.0000
10.73610.37580.24400.14930.04640.01070.00170.00010.0000
20.92980.67780.52560.38280.16730.05470.01230.00160.0001
30.98720.87910.77590.64960.38230.17190.05480.01060.00090.0000
40.99840.96720.92190.84970.63310.37700.16620.04730.00640.0001
50.99990.99360.98030.95270.83380.62300.36690.15030.03280.0016
61.00000.99910.99650.98940.94520.82810.61770.35040.12090.0128
70.99990.99960.99840.98770.94530.83270.61720.32220.0702
81.00001.00000.99990.99830.98930.95360.85070.62420.2639
91.00000.99990.99900.99400.97180.89260.6513
101.00001.00001.00001.00001.00001.0000
1100.31380.08590.04220.01980.00360.00050.0000
10.69740.32210.19710.11300.03020.00590.00070.0000
20.91040.61740.45520.31270.11890.03270.00590.00060.0000
30.98150.83890.71330.56960.29630.11330.02930.00430.0002
40.99720.94960.88540.78970.53280.27440.09940.02160.00200.0000
50.99970.98830.96570.92180.75350.50000.24650.07820.01170.0003
61.00000.99800.99240.97840.90060.72560.46720.21030.05040.0028
70.99980.99880.99570.97070.88670.70370.43040.16110.0185
81.00000.99990.99940.99410.96730.88110.68730.38260.0896
91.00001.00000.99930.99410.96980.88700.67790.3026
101.00000.99950.99640.98020.91410.6862
111.00001.00001.00001.00001.0000
p
nr0.100.200.250.300.400.500.600.700.800.90
1200.28240.06870.03170.01380.00220.00020.0000
10.65900.27490.15840.08500.01960.00320.00030.0000
20.88910.55830.39070.25280.08340.01930.00280.00020.0000
30.97440.79460.64880.49250.22530.07300.01530.00170.0001
40.99570.92740.84240.72370.43820.19380.05730.00950.00060.0000
50.99950.98060.94560.88220.66520.38720.15820.03860.00390.0001
60.99990.99610.98570.96140.84180.61280.33480.11780.01940.0005
71.00000.99940.99720.99050.94270.80620.56180.27630.07260.0043
80.99990.99960.99830.98470.92700.77470.50750.20540.0256
91.00001.00000.99980.99720.98070.91660.74720.44170.1109
101.00000.99970.99680.98040.91500.72510.3410
111.00000.99980.99780.98620.93130.7176
121.00001.00001.00001.00001.0000
1300.25420.05500.02380.00970.00130.00010.0000
10.62130.23360.12670.06370.01260.00170.00010.0000
20.86610.50170.33260.20250.05790.01120.00130.0001
30.96580.74730.58430.42060.16860.04610.00780.00070.0000
40.99350.90090.79400.65430.35300.13340.03210.00400.0002
50.99910.97000.91980.83460.57440.29050.09770.01820.00120.0000
60.99990.99300.97570.93760.77120.50000.22880.06240.00700.0001
71.00000.99880.99440.98180.90230.70950.42560.16540.03000.0009
80.99980.99900.99600.96790.86660.64700.34570.09910.0065
91.00000.99990.99930.99220.95390.83140.57940.25270.0342
101.00000.99990.99870.98880.94210.79750.49830.1339
111.00000.99990.99830.98740.93630.76640.3787
121.00000.99990.99870.99030.94500.7458
131.00001.00001.00001.00001.0000
1400.22880.04400.01780.00680.00080.00010.0000
10.58460.19790.10100.04750.00810.00090.0001
20.84160.44810.28110.16080.03980.00650.00060.0000
30.95590.69820.52130.35520.12430.02870.00390.0002
40.99080.87020.74150.58420.27930.08980.01750.00170.0000
50.99850.95610.88830.78050.48590.21200.05830.00830.0004
60.99980.98840.96170.90670.69250.39530.15010.03150.00240.0000
71.00000.99760.98970.96850.84990.60470.30750.09330.01160.0002
80.99960.99780.99170.94170.78800.51410.21950.04390.0015
91.00000.99970.99830.98250.91020.72070.41580.12980.0092
101.00000.99980.99610.97130.87570.64480.30180.0441
111.00000.99940.99350.96020.83920.55190.1584
120.99990.99910.99190.95250.80210.4154
131.00000.99990.99920.99320.95600.7712
141.00001.00001.00001.00001.0000
730Appendix A Statistical Tables and ProofsrTable A.1 (continued) Binomial Probability Sums b(x; n, p) x=0
Table A.1 Binomial Probability Table731rTable A.1 (continued) Binomial Probability Sums b(x; n, p) x=0
p
nr0.100.200.250.300.400.500.600.700.800.90
1500.20590.03520.01340.00470.00050.0000
10.54900.16710.08020.03530.00520.00050.0000
20.81590.39800.23610.12680.02710.00370.00030.0000
30.94440.64820.46130.29690.09050.01760.00190.0001
40.98730.83580.68650.51550.21730.05920.00930.00070.0000
50.99780.93890.85160.72160.40320.15090.03380.00370.0001
60.99970.98190.94340.86890.60980.30360.09500.01520.0008
71.00000.99580.98270.95000.78690.50000.21310.05000.00420.0000
80.99920.99580.98480.90500.69640.39020.13110.01810.0003
90.99990.99920.99630.96620.84910.59680.27840.06110.0022
101.00000.99990.99930.99070.94080.78270.48450.16420.0127
111.00000.99990.99810.98240.90950.70310.35180.0556
121.00000.99970.99630.97290.87320.60200.1841
131.00000.99950.99480.96470.83290.4510
141.00000.99950.99530.96480.7941
151.00001.00001.00001.0000
1600.18530.02810.01000.00330.00030.0000
10.51470.14070.06350.02610.00330.00030.0000
20.78920.35180.19710.09940.01830.00210.0001
30.93160.59810.40500.24590.06510.01060.00090.0000
40.98300.79820.63020.44990.16660.03840.00490.0003
50.99670.91830.81030.65980.32880.10510.01910.00160.0000
60.99950.97330.92040.82470.52720.22720.05830.00710.0002
70.99990.99300.97290.92560.71610.40180.14230.02570.00150.0000
81.00000.99850.99250.97430.85770.59820.28390.07440.00700.0001
90.99980.99840.99290.94170.77280.47280.17530.02670.0005
101.00000.99970.99840.98090.89490.67120.34020.08170.0033
111.00000.99970.99510.96160.83340.55010.20180.0170
121.00000.99910.98940.93490.75410.40190.0684
130.99990.99790.98170.90060.64820.2108
141.00000.99970.99670.97390.85930.4853
151.00000.99970.99670.97190.8147
161.00001.00001.00001.0000
p
nr0.100.200.250.300.400.500.600.700.800.90
1700.16680.02250.00750.00230.00020.0000
10.48180.11820.05010.01930.00210.00010.0000
20.76180.30960.16370.07740.01230.00120.0001
30.91740.54890.35300.20190.04640.00640.00050.0000
40.97790.75820.57390.38870.12600.02450.00250.0001
50.99530.89430.76530.59680.26390.07170.01060.00070.0000
60.99920.96230.89290.77520.44780.16620.03480.00320.0001
70.99990.98910.95980.89540.64050.31450.09190.01270.0005
81.00000.99740.98760.95970.80110.50000.19890.04030.00260.0000
90.99950.99690.98730.90810.68550.35950.10460.01090.0001
100.99990.99940.99680.96520.83380.55220.22480.03770.0008
111.00000.99990.99930.98940.92830.73610.40320.10570.0047
121.00000.99990.99750.97550.87400.61130.24180.0221
131.00000.99950.99360.95360.79810.45110.0826
140.99990.99880.98770.92260.69040.2382
151.00000.99990.99790.98070.88180.5182
161.00000.99980.99770.97750.8332
171.00001.00001.00001.0000
1800.15010.01800.00560.00160.00010.0000
10.45030.09910.03950.01420.00130.0001
20.73380.27130.13530.06000.00820.00070.0000
30.90180.50100.30570.16460.03280.00380.0002
40.97180.71640.51870.33270.09420.01540.00130.0000
50.99360.86710.71750.53440.20880.04810.00580.0003
60.99880.94870.86100.72170.37430.11890.02030.00140.0000
70.99980.98370.94310.85930.56340.24030.05760.00610.0002
81.00000.99570.98070.94040.73680.40730.13470.02100.0009
90.99910.99460.97900.86530.59270.26320.05960.00430.0000
100.99980.99880.99390.94240.75970.43660.14070.01630.0002
111.00000.99980.99860.97970.88110.62570.27830.05130.0012
121.00000.99970.99420.95190.79120.46560.13290.0064
131.00000.99870.98460.90580.66730.28360.0282
140.99980.99620.96720.83540.49900.0982
151.00000.99930.99180.94000.72870.2662
160.99990.99870.98580.90090.5497
171.00000.99990.99840.98200.8499
181.00001.00001.00001.0000
p
nr0.100.200.250.300.400.500.600.700.800.90
1900.13510.01440.00420.00110.0001
10.42030.08290.03100.01040.00080.0000
20.70540.23690.11130.04620.00550.00040.0000
30.88500.45510.26310.13320.02300.00220.0001
40.96480.67330.46540.28220.06960.00960.00060.0000
50.99140.83690.66780.47390.16290.03180.00310.0001
60.99830.93240.82510.66550.30810.08350.01160.0006
70.99970.97670.92250.81800.48780.17960.03520.00280.0000
81.00000.99330.97130.91610.66750.32380.08850.01050.0003
90.99840.99110.96740.81390.50000.18610.03260.0016
100.99970.99770.98950.91150.67620.33250.08390.00670.0000
111.00000.99950.99720.96480.82040.51220.18200.02330.0003
120.99990.99940.98840.91650.69190.33450.06760.0017
131.00000.99990.99690.96820.83710.52610.16310.0086
141.00000.99940.99040.93040.71780.32670.0352
150.99990.99780.97700.86680.54490.1150
161.00000.99960.99450.95380.76310.2946
171.00000.99920.98960.91710.5797
180.99990.99890.98560.8649
191.00001.00001.00001.0000
2000.12160.01150.00320.00080.0000
10.39170.06920.02430.00760.00050.0000
20.67690.20610.09130.03550.00360.0002
30.86700.41140.22520.10710.01600.00130.0000
40.95680.62960.41480.23750.05100.00590.0003
50.98870.80420.61720.41640.12560.02070.00160.0000
60.99760.91330.78580.60800.25000.05770.00650.0003
70.99960.96790.89820.77230.41590.13160.02100.00130.0000
80.99990.99000.95910.88670.59560.25170.05650.00510.0001
91.00000.99740.98610.95200.75530.41190.12750.01710.0006
100.99940.99610.98290.87250.58810.24470.04800.00260.0000
110.99990.99910.99490.94350.74830.40440.11330.01000.0001
121.00000.99980.99870.97900.86840.58410.22770.03210.0004
131.00000.99970.99350.94230.75000.39200.08670.0024
141.00000.99840.97930.87440.58360.19580.0113
150.99970.99410.94900.76250.37040.0432
161.00000.99870.98400.89290.58860.1330
170.99980.99640.96450.79390.3231
181.00000.99950.99240.93080.6083
191.00000.99920.98850.8784
201.00001.00001.0000
732Appendix A Statistical Tables and Proofs
Table A.2 Poisson Probability Table732rTable A.2 (continued) Poisson Probability Sums p(x; )
Table A.2 Poisson Probability Sums
r p(x; )
x=0
r0.10.20.30.40.50.60.70.80.9
00.90480.81870.74080.67030.60650.54880.49660.44930.4066
10.99530.98250.96310.93840.90980.87810.84420.80880.7725
20.99980.99890.99640.99210.98560.97690.96590.95260.9371
31.00000.99990.99970.99920.99820.99660.99420.99090.9865
41.00001.00000.99990.99980.99960.99920.99860.9977
51.00001.00001.00000.99990.99980.9997
61.00001.00001.0000
r1.01.52.02.53.03.54.04.55.0
00.36790.22310.13530.08210.04980.03020.01830.01110.0067
10.73580.55780.40600.28730.19910.13590.09160.06110.0404
20.91970.80880.67670.54380.42320.32080.23810.17360.1247
30.98100.93440.85710.75760.64720.53660.43350.34230.2650
40.99630.98140.94730.89120.81530.72540.62880.53210.4405
50.99940.99550.98340.95800.91610.85760.78510.70290.6160
60.99990.99910.99550.98580.96650.93470.88930.83110.7622
71.00000.99980.99890.99580.98810.97330.94890.91340.8666
81.00000.99980.99890.99620.99010.97860.95970.9319
91.00000.99970.99890.99670.99190.98290.9682
100.99990.99970.99900.99720.99330.9863
111.00000.99990.99970.99910.99760.9945
121.00000.99990.99970.99920.9980
131.00000.99990.99970.9993
141.00000.99990.9998
151.00000.9999
161.0000
x=0
r5.56.06.57.07.58.08.59.09.5
00.00410.00250.00150.00090.00060.00030.00020.00010.0001
10.02660.01740.01130.00730.00470.00300.00190.00120.0008
20.08840.06200.04300.02960.02030.01380.00930.00620.0042
30.20170.15120.11180.08180.05910.04240.03010.02120.0149
40.35750.28510.22370.17300.13210.09960.07440.05500.0403
50.52890.44570.36900.30070.24140.19120.14960.11570.0885
60.68600.60630.52650.44970.37820.31340.25620.20680.1649
70.80950.74400.67280.59870.52460.45300.38560.32390.2687
80.89440.84720.79160.72910.66200.59250.52310.45570.3918
90.94620.91610.87740.83050.77640.71660.65300.58740.5218
100.97470.95740.93320.90150.86220.81590.76340.70600.6453
110.98900.97990.96610.94670.92080.88810.84870.80300.7520
120.99550.99120.98400.97300.95730.93620.90910.87580.8364
130.99830.99640.99290.98720.97840.96580.94860.92610.8981
140.99940.99860.99700.99430.98970.98270.97260.95850.9400
150.99980.99950.99880.99760.99540.99180.98620.97800.9665
160.99990.99980.99960.99900.99800.99630.99340.98890.9823
171.00000.99990.99980.99960.99920.99840.99700.99470.9911
181.00000.99990.99990.99970.99930.99870.99760.9957
191.00001.00000.99990.99970.99950.99890.9980
200.99990.99980.99960.9991
211.00000.99990.99980.9996
221.00000.99990.9999
231.00000.9999
241.0000
Table A.2 (continued) Poisson Probability Sums
r p(x; )
x=0
r10.011.012.013.014.015.016.017.018.0
00.00000.00000.0000
10.00050.00020.00010.00000.0000
20.00280.00120.00050.00020.00010.00000.0000
30.01030.00490.00230.00110.00050.00020.00010.00000.0000
40.02930.01510.00760.00370.00180.00090.00040.00020.0001
50.06710.03750.02030.01070.00550.00280.00140.00070.0003
60.13010.07860.04580.02590.01420.00760.00400.00210.0010
70.22020.14320.08950.05400.03160.01800.01000.00540.0029
80.33280.23200.15500.09980.06210.03740.02200.01260.0071
90.45790.34050.24240.16580.10940.06990.04330.02610.0154
100.58300.45990.34720.25170.17570.11850.07740.04910.0304
110.69680.57930.46160.35320.26000.18480.12700.08470.0549
120.79160.68870.57600.46310.35850.26760.19310.13500.0917
130.86450.78130.68150.57300.46440.36320.27450.20090.1426
140.91650.85400.77200.67510.57040.46570.36750.28080.2081
150.95130.90740.84440.76360.66940.56810.46670.37150.2867
160.97300.94410.89870.83550.75590.66410.56600.46770.3751
170.98570.96780.93700.89050.82720.74890.65930.56400.4686
180.99280.98230.96260.93020.88260.81950.74230.65500.5622
190.99650.99070.97870.95730.92350.87520.81220.73630.6509
200.99840.99530.98840.97500.95210.91700.86820.80550.7307
210.99930.99770.99390.98590.97120.94690.91080.86150.7991
220.99970.99900.99700.99240.98330.96730.94180.90470.8551
230.99990.99950.99850.99600.99070.98050.96330.93670.8989
241.00000.99980.99930.99800.99500.98880.97770.95940.9317
250.99990.99970.99900.99740.99380.98690.97480.9554
261.00000.99990.99950.99870.99670.99250.98480.9718
270.99990.99980.99940.99830.99590.99120.9827
281.00000.99990.99970.99910.99780.99500.9897
291.00000.99990.99960.99890.99730.9941
300.99990.99980.99940.99860.9967
311.00000.99990.99970.99930.9982
321.00000.99990.99960.9990
330.99990.99980.9995
341.00000.99990.9998
351.00000.9999
360.9999
371.0000
Table A.3 Areas under the Normal Curve 0 z
z.00.01.02.03.04.05.06.07.08.09
3.40.00030.00030.00030.00030.00030.00030.00030.00030.00030.0002
3.30.00050.00050.00050.00040.00040.00040.00040.00040.00040.0003
3.20.00070.00070.00060.00060.00060.00060.00060.00050.00050.0005
3.10.00100.00090.00090.00090.00080.00080.00080.00080.00070.0007
3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010
2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014
2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019
2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026
2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036
2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048
2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064
2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084
2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110
2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143
2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183
1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233
1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294
1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367
1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455
1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559
1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681
1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823
1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985
1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170
1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379
0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611
0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867
0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148
0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451
0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776
0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121
0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483
0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859
0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247
0.00.50000.49600.49200.48800.48400.48010.47610.47210.46810.4641
736Appendix A Statistical Tables and Proofs
Table A.3 Normal Probability Table736Area
Table A.3 (continued) Areas under the Normal Curve
z.00.01.02.03.04.05.06.07.08.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
3.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993
3.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995
3.30.99950.99950.99950.99960.99960.99960.99960.99960.99960.9997
3.40.99970.99970.99970.99970.99970.99970.99970.99970.99970.9998
Table A.4 Critical Values of the t-Distribution
0 t
738Appendix A Statistical Tables and Proofs
Table A.4 Student t-Distribution Probability Table738
v0.400.300.200.150.100.050.025
10.3250.7271.3761.9633.0786.31412.706
20.2890.6171.0611.3861.8862.9204.303
30.2770.5840.9781.2501.6382.3533.182
40.2710.5690.9411.1901.5332.1322.776
50.2670.5590.9201.1561.4762.0152.571
60.2650.5530.9061.1341.4401.9432.447
70.2630.5490.8961.1191.4151.8952.365
80.2620.5460.8891.1081.3971.8602.306
90.2610.5430.8831.1001.3831.8332.262
100.2600.5420.8791.0931.3721.8122.228
110.2600.5400.8761.0881.3631.7962.201
120.2590.5390.8731.0831.3561.7822.179
130.2590.5380.8701.0791.3501.7712.160
140.2580.5370.8681.0761.3451.7612.145
150.2580.5360.8661.0741.3411.7532.131
160.2580.5350.8651.0711.3371.7462.120
170.2570.5340.8631.0691.3331.7402.110
180.2570.5340.8621.0671.3301.7342.101
190.2570.5330.8611.0661.3281.7292.093
200.2570.5330.8601.0641.3251.7252.086
210.2570.5320.8591.0631.3231.7212.080
220.2560.5320.8581.0611.3211.7172.074
230.2560.5320.8581.0601.3191.7142.069
240.2560.5310.8571.0591.3181.7112.064
250.2560.5310.8561.0581.3161.7082.060
260.2560.5310.8561.0581.3151.7062.056
270.2560.5310.8551.0571.3141.7032.052
280.2560.5300.8551.0561.3131.7012.048
290.2560.5300.8541.0551.3111.6992.045
300.2560.5300.8541.0551.3101.6972.042
400.2550.5290.8511.0501.3031.6842.021
600.2540.5270.8481.0451.2961.6712.000
1200.2540.5260.8451.0411.2891.6581.980
0.2530.5240.8421.0361.2821.6451.960
Table A.4 (continued) Critical Values of the t-Distribution
v0.020.0150.010.00750.0050.00250.0005
115.89421.20531.82142.43363.656127.321636.578
24.8495.6436.9658.0739.92514.08931.600
33.4823.8964.5415.0475.8417.45312.924
42.9993.2983.7474.0884.6045.5988.610
52.7573.0033.3653.6344.0324.7736.869
62.6122.8293.1433.3723.7074.3175.959
72.5172.7152.9983.2033.4994.0295.408
82.4492.6342.8963.0853.3553.8335.041
92.3982.5742.8212.9983.2503.6904.781
102.3592.5272.7642.9323.1693.5814.587
112.3282.4912.7182.8793.1063.4974.437
122.3032.4612.6812.8363.0553.4284.318
132.2822.4362.6502.8013.0123.3724.221
142.2642.4152.6242.7712.9773.3264.140
152.2492.3972.6022.7462.9473.2864.073
162.2352.3822.5832.7242.9213.2524.015
172.2242.3682.5672.7062.8983.2223.965
182.2142.3562.5522.6892.8783.1973.922
192.2052.3462.5392.6742.8613.1743.883
202.1972.3362.5282.6612.8453.1533.850
212.1892.3282.5182.6492.8313.1353.819
222.1832.3202.5082.6392.8193.1193.792
232.1772.3132.5002.6292.8073.1043.768
242.1722.3072.4922.6202.7973.0913.745
252.1672.3012.4852.6122.7873.0783.725
262.1622.2962.4792.6052.7793.0673.707
272.1582.2912.4732.5982.7713.0573.689
282.1542.2862.4672.5922.7633.0473.674
292.1502.2822.4622.5862.7563.0383.660
302.1472.2782.4572.5812.7503.0303.646
402.1232.2502.4232.5422.7042.9713.551
602.0992.2232.3902.5042.6602.9153.460
1202.0762.1962.3582.4682.6172.8603.373
2.054 2.170 2.326 2.432 2.576 2.807 3.290
2Table A.5 Critical Values of the Chi-Squared Distribution 0
v0.9950.990.980.9750.950.900.800.750.700.50
10.04 3930.03 1570.03 6280.03 9820.003930.01580.06420.1020.1480.455
20.01000.02010.04040.05060.1030.2110.4460.5750.7131.386
30.07170.1150.1850.2160.3520.5841.0051.2131.4242.366
40.2070.2970.4290.4840.7111.0641.6491.9232.1953.357
50.4120.5540.7520.8311.1451.6102.3432.6753.0004.351
60.6760.8721.1341.2371.6352.2043.0703.4553.8285.348
70.9891.2391.5641.6902.1672.8333.8224.2554.6716.346
81.3441.6472.0322.1802.7333.4904.5945.0715.5277.344
91.7352.0882.5322.7003.3254.1685.3805.8996.3938.343
102.1562.5583.0593.2473.9404.8656.1796.7377.2679.342
112.6033.0533.6093.8164.5755.5786.9897.5848.14810.341
123.0743.5714.1784.4045.2266.3047.8078.4389.03411.340
133.5654.1074.7655.0095.8927.0418.6349.2999.92612.340
144.0754.6605.3685.6296.5717.7909.46710.16510.82113.339
154.6015.2295.9856.2627.2618.54710.30711.03711.72114.339
165.1425.8126.6146.9087.9629.31211.15211.91212.62415.338
175.6976.4087.2557.5648.67210.08512.00212.79213.53116.338
186.2657.0157.9068.2319.39010.86512.85713.67514.44017.338
196.8447.6338.5678.90710.11711.65113.71614.56215.35218.338
207.4348.2609.2379.59110.85112.44314.57815.45216.26619.337
218.0348.8979.91510.28311.59113.24015.44516.34417.18220.337
228.6439.54210.60010.98212.33814.04116.31417.24018.10121.337
239.26010.19611.29311.68913.09114.84817.18718.13719.02122.337
249.88610.85611.99212.40113.84815.65918.06219.03719.94323.337
2510.52011.52412.69713.12014.61116.47318.94019.93920.86724.337
2611.16012.19813.40913.84415.37917.29219.82020.84321.79225.336
2711.80812.87814.12514.57316.15118.11420.70321.74922.71926.336
2812.46113.56514.84715.30816.92818.93921.58822.65723.64727.336
2913.12114.25615.57416.04717.70819.76822.47523.56724.57728.336
3013.78714.95316.30616.79118.49320.59923.36424.47825.50829.336
4020.70722.16423.83824.43326.50929.05132.34533.6634.87239.335
5027.99129.70731.66432.35734.76437.68941.44942.94244.31349.335
6035.53437.48539.69940.48243.18846.45950.64152.29453.80959.335
740Appendix A Statistical Tables and Proofs
Table A.5 Chi-Squared Distribution Probability Table740
Table A.5 (continued) Critical Values of the Chi-Squared Distribution
v0.300.250.200.100.050.0250.020.010.0050.001
11.0741.3231.6422.7063.8415.0245.4126.6357.87910.827
22.4082.7733.2194.6055.9917.3787.8249.21010.59713.815
33.6654.1084.6426.2517.8159.3489.83711.34512.83816.266
44.8785.3855.9897.7799.48811.14311.66813.27714.86018.466
56.0646.6267.2899.23611.07012.83213.38815.08616.75020.515
67.2317.8418.55810.64512.59214.44915.03316.81218.54822.457
78.3839.0379.80312.01714.06716.01316.62218.47520.27824.321
89.52410.21911.03013.36215.50717.53518.16820.09021.95526.124
910.65611.38912.24214.68416.91919.02319.67921.66623.58927.877
1011.78112.54913.44215.98718.30720.48321.16123.20925.18829.588
1112.89913.70114.63117.27519.67521.92022.61824.72526.75731.264
1214.01114.84515.81218.54921.02623.33724.05426.21728.30032.909
1315.11915.98416.98519.81222.36224.73625.47127.68829.81934.527
1416.22217.11718.15121.06423.68526.11926.87329.14131.31936.124
1517.32218.24519.31122.30724.99627.48828.25930.57832.80137.698
1618.41819.36920.46523.54226.29628.84529.63332.00034.26739.252
1719.51120.48921.61524.76927.58730.19130.99533.40935.71840.791
1820.60121.60522.76025.98928.86931.52632.34634.80537.15642.312
1921.68922.71823.90027.20430.14432.85233.68736.19138.58243.819
2022.77523.82825.03828.41231.41034.17035.02037.56639.99745.314
2123.85824.93526.17129.61532.67135.47936.34338.93241.40146.796
2224.93926.03927.30130.81333.92436.78137.65940.28942.79648.268
2326.01827.14128.42932.00735.17238.07638.96841.63844.18149.728
2427.09628.24129.55333.19636.41539.36440.27042.98045.55851.179
2528.17229.33930.67534.38237.65240.64641.56644.31446.92852.619
2629.24630.43531.79535.56338.88541.92342.85645.64248.29054.051
2730.31931.52832.91236.74140.11343.19544.14046.96349.64555.475
2831.39132.62034.02737.91641.33744.46145.41948.27850.99456.892
2932.46133.71135.13939.08742.55745.72246.69349.58852.33558.301
3033.53034.80036.25040.25643.77346.97947.96250.89253.67259.702
4044.16545.61647.26951.80555.75859.34260.43663.69166.76673.403
5054.72356.33458.16463.16767.50571.42072.61376.15479.49086.660
6065.22666.98168.97274.39779.08283.29884.5888.37991.95299.608
Table A.6 Critical Values of the F-Distribution 0 f
f0.05 (v1 , v2 )
v1
v2123456789
1161.45199.50215.71224.58230.16233.99236.77238.88240.54
218.5119.0019.1619.2519.3019.3319.3519.3719.38
310.139.559.289.129.018.948.898.858.81
47.716.946.596.396.266.166.096.046.00
56.615.795.415.195.054.954.884.824.77
65.995.144.764.534.394.284.214.154.10
75.594.744.354.123.973.873.793.733.68
85.324.464.073.843.693.583.503.443.39
95.124.263.863.633.483.373.293.233.18
104.964.103.713.483.333.223.143.073.02
114.843.983.593.363.203.093.012.952.90
124.753.893.493.263.113.002.912.852.80
134.673.813.413.183.032.922.832.772.71
144.603.743.343.112.962.852.762.702.65
154.543.683.293.062.902.792.712.642.59
164.493.633.243.012.852.742.662.592.54
174.453.593.202.962.812.702.612.552.49
184.413.553.162.932.772.662.582.512.46
194.383.523.132.902.742.632.542.482.42
204.353.493.102.872.712.602.512.452.39
214.323.473.072.842.682.572.492.422.37
224.303.443.052.822.662.552.462.402.34
234.283.423.032.802.642.532.442.372.32
244.263.403.012.782.622.512.422.362.30
254.243.392.992.762.602.492.402.342.28
264.233.372.982.742.592.472.392.322.27
274.213.352.962.732.572.462.372.312.25
284.203.342.952.712.562.452.362.292.24
294.183.332.932.702.552.432.352.282.22
304.173.322.922.692.532.422.332.272.21
404.083.232.842.612.452.342.252.182.12
604.003.152.762.532.372.252.172.102.04
1203.923.072.682.452.292.182.092.021.96
3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88Reproduced from Table 18 of Biometrika Tables for Statisticians, Vol. I, by permission of E.S. Pearson and the Biometrika Trustees.
744Appendix A Statistical Tables and ProofsTable A.6 (continued) Critical Values of the F-Distribution
Table A.6 F-Distribution Probability Table744
f0.05 (v1 , v2 )v1 v2 10 12 15 20 24 30 40 60 120 1 241.88 243.91 245.95 248.01 249.05 250.10 251.14 252.20 253.25 254.312 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.503 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.534 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.635 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.366 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.677 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.238 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.939 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.7110 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.5411 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.4012 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.3013 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.2114 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.1315 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.0716 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.0117 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.9618 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.9219 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.8820 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.8421 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.8122 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.7823 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.7624 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.7325 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.7126 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.6927 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.6728 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.6529 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.6430 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.6240 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.5160 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39120 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00
Table A.6 (continued) Critical Values of the F-Distributionf0.01 (v1 , v2 )v1
v2123456789
14052.184999.505403.355624.585763.655858.995928.365981.076022.47
298.5099.0099.1799.2599.3099.3399.3699.3799.39
334.1230.8229.4628.7128.2427.9127.6727.4927.35
421.2018.0016.6915.9815.5215.2114.9814.8014.66
516.2613.2712.0611.3910.9710.6710.4610.2910.16
613.7510.929.789.158.758.478.268.107.98
712.259.558.457.857.467.196.996.846.72
811.268.657.597.016.636.376.186.035.91
910.568.026.996.426.065.805.615.475.35
1010.047.566.555.995.645.395.205.064.94
119.657.216.225.675.325.074.894.744.63
129.336.935.955.415.064.824.644.504.39
139.076.705.745.214.864.624.444.304.19
148.866.515.565.044.694.464.284.144.03
158.686.365.424.894.564.324.144.003.89
168.536.235.294.774.444.204.033.893.78
178.406.115.184.674.344.103.933.793.68
188.296.015.094.584.254.013.843.713.60
198.185.935.014.504.173.943.773.633.52
208.105.854.944.434.103.873.703.563.46
218.025.784.874.374.043.813.643.513.40
227.955.724.824.313.993.763.593.453.35
237.885.664.764.263.943.713.543.413.30
247.825.614.724.223.903.673.503.363.26
257.775.574.684.183.853.633.463.323.22
267.725.534.644.143.823.593.423.293.18
277.685.494.604.113.783.563.393.263.15
287.645.454.574.073.753.533.363.233.12
297.605.424.544.043.733.503.333.203.09
307.565.394.514.023.703.473.303.173.07
407.315.184.313.833.513.293.122.992.89
607.084.984.133.653.343.122.952.822.72
1206.854.793.953.483.172.962.792.662.56
6.634.613.783.323.022.802.642.512.41
f0.01 (v1 , v2 )v1 v2 10 12 15 20 24 30 40 60 120 1 6055.85 6106.32 6157.28 6208.73 6234.63 6260.65 6286.78 6313.03 6339.39 6365.862 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.503 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.134 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.465 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.026 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.887 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.658 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.869 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.3110 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.9111 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.6012 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.3613 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.1714 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.0015 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.8716 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.7517 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.6518 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.5719 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.4920 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.4221 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.3622 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.3123 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.2624 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.2125 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.1726 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.1327 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.1028 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.0629 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.0330 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.0140 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.8060 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60120 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00
745Table A.7 Tolerance Factors for Normal Distributions Table A.7 Tolerance Factors for Normal Distributions Two-Sided Intervals One-Sided Intervals = 0.05 = 0.01 = 0.05 = 0.01 1 1 1 1 n 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.992 32.019 37.674 48.430 160.193 188.491 242.300 20.581 26.260 37.094 103.029 131.426 185.6173 8.380 9.916 12.861 18.930 22.401 29.055 6.156 7.656 10.553 13.995 17.170 23.8964 5.369 6.370 8.299 9.398 11.150 14.527 4.162 5.144 7.042 7.380 9.083 12.3875 4.275 5.079 6.634 6.612 7.855 10.260 3.407 4.203 5.741 5.362 6.578 8.9396 3.712 4.414 5.775 5.337 6.345 8.301 3.006 3.708 5.062 4.411 5.406 7.3357 3.369 4.007 5.248 4.613 5.488 7.187 2.756 3.400 4.642 3.859 4.728 6.4128 3.136 3.732 4.891 4.147 4.936 6.468 2.582 3.187 4.354 3.497 4.285 5.8129 2.967 3.532 4.631 3.822 4.550 5.966 2.454 3.031 4.143 3.241 3.972 5.38910 2.839 3.379 4.433 3.582 4.265 5.594 2.355 2.911 3.981 3.048 3.738 5.07411 2.737 3.259 4.277 3.397 4.045 5.308 2.275 2.815 3.852 2.898 3.556 4.82912 2.655 3.162 4.150 3.250 3.870 5.079 2.210 2.736 3.747 2.777 3.410 4.63313 2.587 3.081 4.044 3.130 3.727 4.893 2.155 2.671 3.659 2.677 3.290 4.47214 2.529 3.012 3.955 3.029 3.608 4.737 2.109 2.615 3.585 2.593 1.189 4.33715 2.480 2.954 3.878 2.945 3.507 4.605 2.068 2.566 3.520 2.522 3.102 4.22216 2.437 2.903 3.812 2.872 3.421 4.492 2.033 2.524 3.464 2.460 3.028 4.12317 2.400 2.858 3.754 2.808 3.345 4.393 2.002 2.486 3.414 2.405 2.963 4.03718 2.366 2.819 3.702 2.753 3.279 4.307 1.974 2.453 3.370 2.357 2.905 3.96019 2.337 2.784 3.656 2.703 3.221 4.230 1.949 2.423 3.331 2.314 2.854 3.89220 2.310 2.752 3.615 2.659 3.168 4.161 1.926 2.396 3.295 2.276 2.808 1.83225 2.208 2.631 3.457 2.494 2.972 3.904 1.838 2.292 3.158 2.129 2.633 3.00130 2.140 2.549 3.350 2.385 2.841 3.733 1.777 2.220 3.064 2.030 2.516 3.44735 2.090 2.490 3.272 2.306 2.748 3.611 1.732 2.167 2.995 1.957 2.430 3.33440 2.052 2.445 3.213 2.247 2.677 3.518 1.697 2.126 2.941 1.902 2.364 3.24945 2.021 2.408 3.165 2.200 2.621 3.444 1.669 2.092 2.898 1.857 2.312 3.18050 1.996 2.379 3.126 2.162 2.576 3.385 1.646 2.065 2.863 1.821 2.269 3.12560 1.958 2.333 3.066 2.103 2.506 3.293 1.609 2.022 2.807 1.764 2.202 3.03870 1.929 2.299 3.021 2.060 2.454 3.225 1.581 1.990 2.765 1.722 2.153 2.97480 1.907 2.272 2.986 2.026 2.414 3.173 1.559 1.965 2.733 1.688 2.114 2.92490 1.889 2.251 2.958 1.999 2.382 3.130 1.542 1.944 2.706 1.661 2.082 2.883100 1.874 2.233 2.934 1.977 2.355 3.096 1.527 1.927 2.684 1.639 2.056 2.850150 1.825 2.175 2.859 1.905 2.270 2.983 1.478 1.870 2.611 1.566 1.971 2.741200 1.798 2.143 2.816 1.865 2.222 2.921 1.450 1.837 2.570 1.524 1.923 2.679250 1.780 2.121 2.788 1.839 2.191 2.880 1.431 1.815 2.542 1.496 1.891 2.638300 1.767 2.106 2.767 1.820 2.169 2.850 1.417 1.800 2.522 1.476 1.868 2.608 1.645 1.960 2.576 1.645 1.960 2.576 1.282 1.645 2.326 1.282 1.645 2.326Adapted from C. Eisenhart, M. W. Hastay, and W. A. Wallis, Techniques of Statistical Analysis, Chapter 2, McGraw-Hill Book Company, New York, 1947. Used with permission of McGraw-Hill Book Company.
746Appendix A Statistical Tables and Proofs
746Appendix A Statistical Tables and Proofs
Table A.8 Sample Size for the t-Test of the Mean
Level of t-Test
Single-Sided TestDouble-Sided Test = 0.005 = 0.01 = 0.01 = 0.02 = 0.025 = 0.05 = 0.05 = 0.1
= 0.1.01 .05.1.2.5 .01 .05.1.2.5 .01 .05.1.2 .5 .01 .05.1.2.5
0.05
0.10
0.15122
0.201399970
0.2511090128 6413910145
0.3013478115631199045122977132
0.351259958109854710988673490725224
0.401159777451018566371178468512610170554019
0.45927762371108168533093675441218055443315
0.5010075635130906655432576544434186545362713
0.558363534226755546362163453728155438302211
0.60715345362263473931185338322413463226199
0.65614639312055413427164633272112392822178
0.70534034281747353024144029241910342419158
0.7547363025164231272113352621169302117137
0.8041322722143728241912312219159271915126
0.8537292420133325211711282117138241714116
0.9034262218122923191610251916127211513105
Value of0.95312420171127211814923171411719141195
= ||/1.00282219161025191613921161310618131185
1.124191614921161412818131196151197
1.221161412818141210715121085131086
1.31815131181613119614109711876
1.4161312107141110961298710875
1.515121197131098611876976
1.613111086121097510876866
1.71210986119879765865
1.812109861087787676
1.91198761087686675
2.010887597767656
2.1108778766766
2.298768765766
2.39776866655
2.487767666
2.587667666
3.076656555
3.56555
4.06
Reproduced with permission from O. L. Davies, ed., Design and Analysis of Industrial Experi- ments, Oliver & Boyd, Edinburgh, 1956.
Table A.9 Table of Sample Sizes for the Test of the Dierence between Two Means 747
Table A.9 Sample Size for the t-Test of the Dierence between Two MeansLevel of t-Test
Single-Sided TestDouble-Sided Test = 0.005 = 0.01 = 0.01 = 0.02 = 0.025 = 0.05 = 0.05 = 0.1
= 0.1.01 .05.1.2.5 .01 .05.1.2.5 .01 .05.1.2.5 .01 .05.1.2.5
0.05
0.10
0.15
0.20137
0.2512488
0.301238761
0.35110906410245
0.408570100501087835
0.4511868101551057939 108866228
0.5096551068245 106866432 88705123
0.551017946 106886838 87715327 112 73584219
0.60101856739 90745832 104 74604523 89 61493616
0.6587735734 104 77644927 88 63513920 76 52423014
0.7010075635029906655432476554434176645362612
0.758866554426795848382167483929155740322311
0.807758493923705143331959423426145035282110
0.85695143352162463830175237312312453125189
0.90624639311955413427154734272111402822168
Value of0.95554235281750373124144230251910362520157
= ||/1.0050383226154533282213382723179332318147
1.142322722133828231911322319148271915126
1.23627231811322420169272016127231613105
1.3312320161028211714823171411620141195
1.427201714924181512820151210617121084
1.5241815138211614117181311951511974
1.6211614117191412106161210851410864
1.7191513107171311961411974129763
1.817137110615121085131086411875
1.916121196141198512976410765
2.01411108613109751187649764
2.113109851298751086538654
2.2121087511976497658654
2.311987510876497657554
2.411986510876486547544
2.51087649765486546543
3.086654765436544543
3.5655436544544343
4.0654454434434
Reproduced with permission from O. L. Davies, ed., Design and Analysis of Industrial Experi- ments, Oliver & Boyd, Edinburgh, 1956.
Table A.10 Critical Values for Bartletts Testbk (0.01; n)Number of Populations, k
n2345678910
30.14110.1672
40.28430.31650.34750.37290.39370.4110
50.39840.43040.46070.48500.50460.52070.53430.54580.5558
60.48500.51490.54300.56530.5832O.59780.61000.62040.6293
70.55120.57870.60450.62480.64100.65420.66520.67440.6824
80.60310.62820.65180.67040.68510.69700.70690.71530.7225
90.64450.66760.68920.70620.71970.73050.73950.74710.7536
100.67830.69960.71950.73520.74750.75750.76570.77260.7786
110.70630.72600.74450.75900.77030.77950.78710.79350.7990
120.72990.74830.76540.77890.78940.79800.80500.81090.8160
130.75010.76720.78320.79580.80560.81350.82010.82560.8303
140.76740.78350.79850.81030.81950.82690.83300.83820.8426
150.78250.79770.81180.82290.83150.83850.84430.84910.8532
160.79580.81010.82350.83390.84210.84860.85410.85860.8625
170.80760.82110.83380.84360.85140.85760.86270.86700.8707
180.81810.83090.84290.85230.85960.86550.87040.87450.8780
190.82750.83970.85120.86010.86700.87270.87730.88110.8845
200.83600.84760.85860.86710.87370.87910.88350.88710.8903
210.84370.85480.86530.87340.87970.88480.88900.89260.8956
220.85070.86140.87140.87910.88520.89010.89410.89750.9004
230.85710.86730.87690.88440.89020.89490.89880.90200.9047
240.86300.87280.88200.88920.89480.89930.90300.90610.9087
250.86840.87790.88670.89360.89900.90340.90690.90990.9124
260.87340.88250.89110.89770.90290.90710.91050.91340.9158
270.87810.88690.89510.90150.90650.91050.91380.91660.9190
280.88240.89090.89880.90500.90990.91380.91690.91960.9219
290.88640.89460.90230.90830.91300.91670.91980.92240.9246
300.89020.89810.90560.91140.91590.91950.92250.92500.9271
400.91750.92350.92910.93350.93700.93970.94200.94390.9455
500.93390.93870.94330.94680.94960.95180.95360.95510.9564
600.94490.94890.95270.95570.95800.95990.96140.96260.9637
800.95860.96170.96460.96680.96850.96990.97110.97200.9728
1000.96690.96930.97160.97340.97480.97590.97690.97760.9783
Reproduced from D. D. Dyer and J. P. Keating, On the Determination of Critical Values forBartletts Test, J. Am. Stat. Assoc., 75, 1980, by permission of the Board of Directors.
748Appendix A Statistical Tables and Proofs
748Appendix A Statistical Tables and Proofs
Table A.10 Table for Bartletts Test 749
Table A.10 (continued) Critical Values for Bartletts Testbk (0.05; n)Number of Populations, k
n2345678910
30.31230.30580.31730.3299
40.47800.46990.48030.49210.50280.51220.52040.52770.5341
50.58450.57620.58500.59520.60450.61260.61970.62600.6315
60.65630.64830.65590.66460.67270.67980.68600.69140.6961
70.70750.70000.70650.71420.72130.72750.73290.73760.7418
80.74560.73870.74440.75120.75740.76290.76770.77190.7757
90.77510.76860.77370.77980.78540.79030.79460.79840.8017
100.79840.79240.79700.80250.80760.81210.81600.81940.8224
110.81750.81180.81600.82100.82570.82980.83330.83650.8392
120.83320.82800.83170.83640.84070.84440.84770.85060.8531
130.84650.84150.84500.84930.85330.85680.85980.86250.8648
140.85780.85320.85640.86040.86410.86730.87010.87260.8748
150.86760.86320.86620.86990.87340.87640.87900.88140.8834
160.87610.87190.87470.87820.88150.88430.88680.88900.8909
170.88360.87960.88230.88560.88860.89130.89360.89570.8975
180.89020.88650.88900.89210.89490.89750.89970.90160.9033
190.89610.89260.89490.89790.90060.90300.90510.90690.9086
200.90150.89800.90030.90310.90570.90800.91000.91170.9132
210.90630.90300.90510.90780.91030.91240.91430.91600.9175
220.91060.90750.90950.91200.91440.91650.91830.91990.9213
230.91460.91160.91350.91590.91820.92020.92190.92350.9248
240.91820.91530.91720.91950.92170.92360.92530.92670.9280
250.92160.91870.92050.92280.92490.92670.92830.92970.9309
260.92460.92190.92360.92580.92780.92960.93110.93250.9336
270.92750.92490.92650.92860.93050.93220.93370.93500.9361
280.93010.92760.92920.93120.93300.93470.93610.93740.9385
290.93260.93010.93160.93360.93540.93700.93830.93960.9406
300.93480.93250.93400.93580.93760.93910.94040.94160.9426
400.95130.94950.95060.95200.95330.95450.95550.95640.9572
500.96120.95970.96060.96170.96280.96370.96450.96520.9658
600.96770.96650.96720.96810.96900.96980.97050.97100.9716
800.97580.97490.97540.97610.97680.97740.97790.97830.9787
1000.98070.97990.98040.98090.98150.98190.98230.98270.9830
Table A.11 Critical Values for Cochrans Test
= 0.01n
Appendix A Statistical Tables and Proofs750 k 2 3 4 5 6 7 8 9 10 11 17 37 145 2 0.9999 0.9950 0.9794 0.9586 0.9373 0.9172 0.8988 0.8823 0.8674 0.8539 0.7949 0.7067 0.6062 0.50003 0.9933 0.9423 0.8831 0.8335 0.7933 0.7606 0.7335 0.7107 0.6912 0.6743 0.6059 0.5153 0.4230 0.33334 0.9676 0.8643 0.7814 0.7212 0.6761 0.6410 0.6129 0.5897 0.5702 0.5536 0.4884 0.4057 0.3251 0.25005 0.9279 0.7885 0.6957 0.6329 0.5875 0.5531 0.5259 0.5037 0.4854 0.4697 0.4094 0.3351 0.2644 0.20006 0.8828 0.7218 0.6258 0.5635 0.5195 0.4866 0.4608 0.4401 0.4229 0.4084 0.3529 0.2858 0.2229 0.16677 0.8376 0.6644 0.5685 0.5080 0.4659 0.4347 0.4105 0.3911 0.3751 0.3616 0.3105 0.2494 0.1929 0.14298 0.7945 0.6152 0.5209 0.4627 0.4226 0.3932 0.3704 0.3522 0.3373 0.3248 0.2779 0.2214 0.1700 0.12509 0.7544 0.5727 0.4810 0.4251 0.3870 0.3592 0.3378 0.3207 0.3067 0.2950 0.2514 0.1992 0.1521 0.111110 0.7175 0.5358 0.4469 0.3934 0.3572 0.3308 0.3106 0.2945 0.2813 0.2704 0.2297 0.1811 0.1376 0.100012 0.6528 0.4751 0.3919 0.3428 0.3099 0.2861 0.2680 0.2535 0.2419 0.2320 0.1961 0.1535 0.1157 0.083315 0.5747 0.4069 0.3317 0.2882 0.2593 0.2386 0.2228 0.2104 0.2002 0.1918 0.1612 0.1251 0.0934 0.066720 0.4799 0.3297 0.2654 0.2288 0.2048 0.1877 0.1748 0.1646 0.1567 0.1501 0.1248 0.0960 0.0709 0.050024 0.4247 0.2871 0.2295 0.1970 0.1759 0.1608 0.1495 0.1406 0.1338 0.1283 0.1060 0.0810 0.0595 0.041730 0.3632 0.2412 0.1913 0.1635 0.1454 0.1327 0.1232 0.1157 0.1100 0.1054 0.0867 0.0658 0.0480 0.033340 0.2940 0.1915 0.1508 0.1281 0.1135 0.1033 0.0957 0.0898 0.0853 0.0816 0.0668 0.0503 0.0363 0.025060 0.2151 0.1371 0.1069 0.0902 0.0796 0.0722 0.0668 0.0625 0.0594 0.0567 0.0461 0.0344 0.0245 0.0167120 0.1225 0.0759 0.0585 0.0489 0.0429 0.0387 0.0357 0.0334 0.0316 0.0302 0.0242 0.0178 0.0125 0.0083 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Reproduced from C. Eisenhart, M. W. Hastay, and W. A. Wallis, Techniques of Statistical Analysis, Chapter 15, McGraw- Hill Book Company, New, York, 1947. Used with permission of McGraw-Hill Book Company.
751Table A.11 Table for Cochrans TestTable A.11 (continued) Critical Values for Cochrans Test = 0.05n k 2 3 4 5 6 7 8 9 10 11 17 37 145 2 0.9985 0.9750 0.9392 0.9057 0.8772 0.8534 0.8332 0.8159 0.8010 0.7880 0.7341 0.6602 0.5813 0.50003 0.9669 0.8709 0.7977 0.7457 0.7071 0.6771 0.6530 0.6333 0.6167 0.6025 0.5466 0.4748 0.4031 0.33334 0.9065 0.7679 0.6841 0.6287 0.5895 0.5598 0.5365 0.5175 0.5017 0.4884 0.4366 0.3720 0.3093 0.25005 0.8412 0.6838 0.5981 0.5441 0.5065 0.4783 0.4564 0.4387 0.4241 0.4118 0.3645 0.3066 0.2513 0.20006 0.7808 0.6161 0.5321 0.4803 0.4447 0.4184 0.3980 0.3817 0.3682 0.3568 0.3135 0.2612 0.2119 0.16677 0.7271 0.5612 0.4800 0.4307 0.3974 0.3726 0.3535 0.3384 0.3259 0.3154 0.2756 0.2278 0.1833 0.14298 0.6798 0.5157 0.4377 0.3910 0.3595 0.3362 0.3185 0.3043 0.2926 0.2829 0.2462 0.2022 0.1616 0.12509 0.6385 0.4775 0.4027 0.3584 0.3286 0.3067 0.2901 0.2768 0.2659 0.2568 0.2226 0.1820 0.1446 0.111110 6.6020 0.4450 0.3733 0.3311 0.3029 0.2823 0.2666 0.2541 0.2439 0.2353 0.2032 0.1655 0.1308 0.100012 0.5410 0.3924 0.3264 0.2880 0.2624 0.2439 0.2299 0.2187 0.2098 0.2020 0.1737 0.1403 0.1100 0.083315 0.4709 0.3346 0.2758 0.2419 0.2195 0.2034 0.1911 0.1815 0.1736 0.1671 0.1429 0.1144 0.0889 0.066720 0.3894 0.2705 0.2205 0.1921 0.1735 0.1602 0.1501 0.1422 0.1357 0.1303 0.1108 0.0879 0.0675 0.050024 0.3434 0.2354 0.1907 0.1656 0.1493 0.1374 0.1286 0.1216 0.1160 0.1113 0.0942 0.0743 0.0567 0.041730 0.2929 0.1980 0.1593 0.1377 0.1237 0.1137 0.1061 0.1002 0.0958 0.0921 0.0771 0.0604 0.0457 0.033340 0.2370 0.1576 0.1259 0.1082 0.0968 0.0887 0.0827 0.0780 0.0745 0.0713 0.0595 0.0462 0.0347 0.025060 0.1737 0.1131 0.0895 0.0765 0.0682 0.0623 0.0583 0.0552 0.0520 0.0497 0.0411 0.0316 0.0234 0.0167120 0.0998 0.0632 0.0495 0.0419 0.0371 0.0337 0.0312 0.0292 0.0279 0.0266 0.0218 0.0165 0.0120 0.0083 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Table A.12 Upper Percentage Points of the Studentized Range Distribution: Values ofq(0.05; k, v)
Degrees ofNumberof Treatmentsk
Freedom, v2345678910
118.027.032.837.240.543.115.147.149.1
26.095.339.8010.8911.7312.4313.0313.5413.99
34.505.916.837.518.048.478.859.189.46
43.935.045.766.296.717.067.357.607.83
53.644.605.225.676.036.336.586.806.99
63.464.344.905.315.635.896.126.326.49
73.344.164.685.065.355.595.805.996.15
83.264.044.534.895.175.405.605.775.92
93.203.954.424.765.025.245.435.605.74
103.153.884.334.664.915.125.305.465.60
113.113.824.264.584.825.035.205.355.49
123.083.774.204.514.754.955.125.275.40
133.063.734.154.464.694.885.055.195.32
143.033.704.114.414.654.834.995.135.25
153.013.674.084.374.594.784.945.085.20
163.003.654.054.344.564.744.905.035.05
172.983.624.024.314.524.704.864.995.11
182.973.614.004.284.494.674.834.965.07
192.963.593.984.264.474.644.794.925.04
202.953.583.964.244.454.624.774.905.01
242.923.533.904.174.374.544.684.814.92
302.893.483.844.114.304.464.604.724.83
402.863.443.794.044.234.394.524.634.74
602.833.403.743.984.164.314.444.554.65
1202.803.363.693.924.104.244.364.474.56
2.77 3.32 3.63 3.86 4.03 4.17 4.29 4.39 4.47
752Appendix A Statistical Tables and Proofs
752Appendix A Statistical Tables and Proofs
Table A.13 Table for Duncans Test 753
Table A.13 Least Signicant Studentized Ranges rp (0.05; p, v) = 0.05
p
v2345678910
117.9717.9717.9717.9717.9717.9717.9717.9717.97
26.0856.0856.0856.0856.0856.0856.0856.0856.085
34.5014.5164.5164.5164.5164.5164.5164.5164.516
43.9274.0134.0334.0334.0334.0334.0334.0334.033
53.6353.7493.7973.8143.8143.8143.8143.8143.814
63.4613.5873.6493.683.6943.6973.6973.6973.697
73.3443.4773.5483.5883.6113.6223.6263.6263.626
83.2613.3993.4753.5213.5493.5663.5753.5793.579
93.1993.3393.4203.4703.5023.5233.5363.5443.547
103.1513.2933.3763.4303.4653.4893.5053.5163.522
113.1133.2563.3423.3973.4353.4623.483.4933.501
123.0823.2253.3133.3703.4103.4393.4593.4743.484
133.0553.2003.2893.3483.3893.4193.4423.4583.470
143.0333.1783.2683.3293.3723.4033.4263.4443.457
153.0143.1603.253.3123.3563.3893.4133.4323.446
162.9983.1443.2353.2983.3433.3763.4023.4223.437
172.9843.1303.2223.2853.3313.3663.3923.4123.429
182.9713.1183.2103.2743.3213.3563.3833.4053.421
192.9603.1073.1993.2643.3113.3473.3753.3973.415
202.9503.0973.1903.2553.3033.3393.3683.3913.409
242.9193.0663.1603.2263.2763.3153.3453.3703.390
302.8883.0353.1313.1993.2503.2903.3223.3493.371
402.8583.0063.1023.1713.2243.2663.3003.3283.352
602.8292.9763.0733.1433.1983.2413.2773.3073.333
1202.8002.9473.0453.1163.1723.2173.2543.2873.314
2.772 2.918 3.017 3.089 3.146 3.193 3.232 3.265 3.294 Abridged from H. L. Harter, Critical Values for Duncans New Multiple Range Test,Biometrics, 16, No. 4, 1960, by permission of the author and the editor.
Table A.13 (continued) Least Signicant Studentized Ranges rp (0.01; p, v) = 0.01
p
v2345678910
190.0390.0390.0390.0390.0390.0390.0390.0390.03
214.0414.0414.0414.0414.0414.0414.0414.0414.04
38.2618.3218.3218.3218.3218.3218.3218.3218.321
46.5126.6776.7406.7566.7566.7566.7566.7566.756
55.7025.8935.9896.0406.0656.0746.0746.0746.074
65.2435.4395.5495.6145.6555.6805.6945.7015.703
74.9495.1455.2605.3345.3835.4165.4395.4545.464
84.7464.9395.0575.1355.1895.2275.2565.2765.291
94.5964.7874.9064.9865.0435.0865.1185.1425.160
104.4824.6714.7904.8714.9314.9755.0105.0375.058
114.3924.5794.6974.7804.8414.8874.9244.9524.975
124.3204.5044.6224.7064.7674.8154.8524.8834.907
134.2604.4424.5604.6444.7064.7554.7934.8244.850
144.2104.3914.5084.5914.6544.7044.7434.7754.802
154.1684.3474.4634.5474.6104.6604.7004.7334.760
164.1314.3094.4254.5094.5724.6224.6634.6964.724
174.0994.2754.3914.4754.5394.5894.6304.6644.693
184.0714.2464.3624.4454.5094.5604.6014.6354.664
194.0464.2204.3354.4194.4834.5344.5754.6104.639
204.0244.1974.3124.3954.4594.5104.5524.5874.617
243.9564.1264.2394.3224.3864.4374.4804.5164.546
303.8894.0564.1684.2504.3144.3664.4094.4454.477
403.8253.9884.0984.1804.2444.2964.3394.3764.408
603.7623.9224.0314.1114.1744.2264.2704.3074.340
1203.7023.8583.9654.0444.1074.1584.2024.2394.272
3.643 3.796 3.900 3.978 4.040 4.091 4.135 4.172 4.205
754Appendix A Statistical Tables and Proofs
754Appendix A Statistical Tables and Proofs
Table A.14 Table for Dunnetts Two-Sided Test 755
Table A.14 Values of d/2 (k, v) for Two-Sided Comparisons between k Treatments and a Control = 0.05k = Number of Treatment Means (excluding control)
v123456789
52.573.033.293.483.623.733.823.903.97
62.452.863.103.263.393.493.573.643.71
72.362.752.973.123.243.333.413.473.53
82.312.672.883.023.133.223.293.353.41
92.262.612.812.953.053.143.203.263.32
102.232.572.762.892.993.073.143.193.24
112.202.532.722.842.943.023.083.143.19
122.182.502.682.812.902.983.043.093.14
132.162.482.652.782.872.943.003.063.10
142.142.462.632.752.842.912.973.023.07
152.132.442.612.732.822.892.953.003.04
162.122.422.592.712.802.872.922.973.02
172.112.412.582.692.782.852.902.953.00
182.102.402.562.682.762.832.892.942.98
192.092.392.552.662.752.812.872.922.96
202.092.382.542.652.732.802.862.902.95
242.062.352.512.612.702.762.812.862.90
302.042.322.472.582.662.722.772.822.86
402.022.292.442.542.622.682.732.772.81
602.002.272.412.512.582.642.692.732.77
1201.982.242.382.472.552.602.652.692.73
1.96 2.21 2.35 2.44 2.51 2.57 2.61 2.65 2.69Reproduced from Charles W. Dunnett, New Tables for Multiple Comparison with a Con- trol, Biometrics, 20, No. 3, 1964, by permission of the author and the editor.
Table A.14 (continued) Values of d/2 (k, v) for Two-Sided Comparisons between k Treat- ments and a Control = 0.01k = Number of Treatment Means (excluding control)
v123456789
54.034.634.985.225.415.565.695.805.89
63.714.214.514.714.875.005.105.205.28
73.503.954.214.394.534.644.744.824.89
83.363.774.004.174.294.404.484.564.62
93.253.633.854.014.124.224.304.374.43
103.173.533.743.883.994.084.164.224.28
113.113.453.653.793.893.984.054.114.16
123.053.393.583.713.813.893.964.024.07
133.013.333.523.653.743.823.893.943.99
142.983.293.473.593.693.763.833.883.93
152.953.253.433.553.643.713.783.833.88
162.923.223.393.513.603.673.733.783.83
172.903.193.363.473.563.633.693.743.79
182.883.173.333.443.533.603.663.713.75
192.863.153.313.423.503.573.633.683.72
202.853.133.293.403.483.553.603.653.69
242.803.073.223.323.403.473.523.573.61
302.753.013.153.253.333.393.443.493.52
402.702.953.093.193.263.323.373.413.44
602.662.903.033.123.193.253.293.333.37
1202.622.852.973.063.123.183.223.263.29
2.58 2.79 2.92 3.00 3.06 3.11 3.15 3.19 3.22
756Appendix A Statistical Tables and Proofs
756Appendix A Statistical Tables and Proofs
Table A.15 Table for Dunnetts One-Sided Test 757
Table A.15 Values of d (k, v) for One-Sided Comparisons between k Treatments and a Control = 0.05k = Number of Treatment Means (excluding control)
v123456789
52.022.442.682.852.983.083.163.243.30
61.942.342.562.712.832.923.003.073.12
71.892.272.482.622.732.822.892.953.01
81.862.222.422.552.662.742.812.872.92
91.832.182.372.502.602.682.752.812.86
101.812.152.342.472.562.642.702.762.81
111.802.132.312.442.532.602.672.722.77
121.782.112.292.412.502.582.642.692.74
131.772.092.272.392.482.552.612.662.71
141.762.082.252.372.462.532.592.642.69
151.752.072.242.362.442.512.572.622.67
161.752.062.232.342.432.502.562.612.65
171.742.052.222.332.422.492.542.592.64
181.732.042.212.322.412.482.532.582.62
191.732.032.202.312.402.472.522.572.61
201.722.032.192.302.392.462.512.562.60
241.712.012.172.282.362.432.482.532.57
301.701.992.152.252.332.402.452.502.54
401.681.972.132.232.312.372.422.472.51
601.671.952.102.212.282.352.392.442.48
1201.661.932.082.182.262.322.372.412.45
1.64 1.92 2.06 2.16 2.23 2.29 2.34 2.38 2.42Reproduced from Charles W. Dunnett, A Multiple Comparison Procedure for Compar- ing Several Treatments with a Control, J. Am. Stat. Assoc., 50, 1955, 10961121, by permission of the author and the editor.
Table A.15 (continued) Values of d (k, v) for One-Sided Comparisons between k Treat- ments and a Control = 0.01k = Number of Treatment Means (excluding control)
v123456789
53.373.904.214.434.604.734.854.945.03
63.143.613.884.074.214.334.434.514.59
73.003.423.663.833.964.074.154.234.30
82.903.293.513.673.793.883.964.034.09
92.823.193.403.553.663.753.823.893.94
102.763.113.313.453.563.643.713.783.83
112.723.063.253.383.483.563.633.693.74
122.683.013.193.323.423.503.563.623.67
132.652.973.153.273.373.443.513.563.61
142.622.943.113.233.323.403.463.513.56
152.602.913.083.203.293.363.423.473.52
162.582.883.053.173.263.333.393.443.48
172.572.863.033.143.233.303.363.413.45
182.552.843.013.123.213.273.333.383.42
192.542.832.993.103.183.253.313.363.40
202.532.812.973.083.173.233.293.343.38
242.492.772.923.033.113.173.223.273.31
302.462.722.872.973.053.113.163.213.24
402.422.682.822.922.993.053.103.143.18
602.392.642.782.872.943.003.043.083.12
1202.362.602.732.822.892.942.993.033.06
2.33 2.56 2.68 2.77 2.84 2.89 2.93 2.97 3.00
758Appendix A Statistical Tables and Proofs
758Appendix A Statistical Tables and Proofs
Table A.16 Table for the Signed-Rank Test 759
Table A.16 Critical Values for the Signed-Rank Test
nOne-Sided = 0.01Two-Sided = 0.02One-Sided = 0.025Two-Sided = 0.05One-Sided = 0.05Two-Sided = 0.1
51
612
7024
8246
9368
105811
1171114
12101417
13131721
14162126
15202530
16243036
17283541
18334047
19384654
20435260
21495968
22566675
23627383
24698192
257790101
268598110
2793107120
28102117130
29111127141
30120137152
Reproduced from F. Wilcoxon and R. A. Wilcox, Some Rapid Approximate Statistical Procedures, American Cyanamid Company, Pearl River, N.Y., 1964, by permission of the American Cyanamid Company.
Table A.17 Critical Values for the Wilcoxon Rank-Sum TestOne-Tailed Test at = 0.001 or Two-Tailed Test at = 0.002n2
n167891011121314151617181920
1
2
30000
400011122333
500112233455677
6012234456789101112
723356789101113141516
8556891112141517182021
97810121415171921232526
101012141719212325272932
1115172022242729323437
12202325283134374042
132629323538424548
1432363943465054
15404347515559
164852566065
1757616670
18667176
197782
20 88
One-Tailed Test at = 0.01 or Two-Tailed Test at = 0.02
n2
n1567891011121314151617181920
1
200000011
300111222334445
401123345567789910
512345678910111213141516
6346789111213151618192022
76891112141617192123242628
810111315172022242628303234
9141618212326283133363840
101922242730333638414447
1125283134374144475053
12313538424649535660
133943475155596367
1447515660656973
15566166707580
166671768287
1777828893
188894100
19101107
20 114Based in part on Tables 1, 3, 5, and 7 of D. Auble, Extended Tables for the Mann-Whitney Statistic, Bulletin of the Institute of Educational Research at Indiana University, 1, No. 2, 1953, by permission of the director.
760Appendix A Statistical Tables and Proofs
760Appendix A Statistical Tables and Proofs
Table A.17 Table for the Rank-Sum Test 761
Table A.17 (continued) Critical Values for the Wilcoxon Rank-Sum TestOne-Tailed Test at = 0.025 or Two-Tailed Test at = 0.05n2
n14567891011121314151617181920
1
20000111112222
30112233445566778
401234456789101111121313
52356789111213141517181920
6568101113141617192122242527
7810121416182022242628303234
813151719222426293134363841
9172023262831343739424548
102326293336394245485255
1130333740444751555862
12374145495357616569
134550545963677276
1455596467747883
15647075808590
167581869298
17879399105
1899106112
19113119
20 127
One-Tailed Test at = 0.05 or Two-Tailed Test at = 0.1n2
n134567891011121314151617181920
100
20001111223333444
300122344556778991011
41234567891011121415161718
5456891112131516181920222325
67810121416171921232526283032
71113151719212426283033353739
815182023262831333639414447
9212427303336394245485154
102731343741444851555862
1134384246505457616569
12424751556064687277
135156616570758084
1461667177828792
157277838894100
16838995101107
1796102109115
18109116123
19123130
20 138
Table A.18 P (V v when H0 is true) in the Runs Test v
(n1 , n2 )2345678910
(2, 3)0.2000.5000.9001.000
(2, 4)0.1330.4000.8001.000
(2, 5)0.0950.3330.7141.000
(2, 6)0.0710.2860.6431.000
(2, 7)0.0560.2500.5831.000
(2, 8)0.0440.2220.5331.000
(2, 9)0.0360.2000.4911.000
(2, 10)0.0300.1820.4551.000
(3, 3)0.1000.3000.7000.9001.000
(3, 4)0.0570.2000.5430.8000.9711.000
(3, 5)0.0360.1430.4290.7140.9291.000
(3, 6)0.0240.1070.3450.6430.8811.000
(3, 7)0.0170.0830.2830.5830.8331.000
(3, 8)0.0120.0670.2360.5330.7881.000
(3, 9)0.0090.0550.2000.4910.7451.000
(3, 10)0.0070.0450.1710.4550.7061.000
(4, 4)0.0290.1140.3710.6290.8860.9711.000
(4, 5)0.0160.0710.2620.5000.7860.9290.9921.000
(4, 6)0.0100.0480.1900.4050.6900.8810.9761.000
(4, 7)0.0060.0330.1420.3330.6060.8330.9541.000
(4, 8)0.0040.0240.1090.2790.5330.7880.9291.000
(4, 9)0.0030.0180.0850.2360.4710.7450.9021.000
(4, 10)0.0020.0140.0680.2030.4190.7060.8741.000
(5, 5)0.0080.0400.1670.3570.6430.8330.9600.9921.000
(5, 6)0.0040.0240.1100.2620.5220.7380.9110.9760.998
(5, 7)0.0030.0150.0760.1970.4240.6520.8540.9550.992
(5, 8)0.0020.0100.0540.1520.3470.5760.7930.9290.984
(5, 9)0.0010.0070.0390.1190.2870.5100.7340.9020.972
(5, 10)0.0010.0050.0290.0950.2390.4550.6780.8740.958
(6, 6)0.0020.0130.0670.1750.3920.6080.8250.9330.987
(6, 7)0.0010.0080.0430.1210.2960.5000.7330.8790.966
(6, 8)0.0010.0050.0280.0860.2260.4130.6460.8210.937
(6, 9)0.0000.0030.0190.0630.1750.3430.5660.7620.902
(6, 10)0.0000.0020.0130.0470.1370.2880.4970.7060.864
(7, 7)0.0010.0040.0250.0780.2090.3830.6170.7910.922
(7, 8)0.0000.0020.0150.0510.1490.2960.5140.7040.867
(7, 9)0.0000.0010.0100.0350.1080.2310.4270.6220.806
(7, 10)0.0000.0010.0060.0240.0800.1820.3550.5490.743
(8, 8)0.0000.0010.0090.0320.1000.2140.4050.5950.786
(8, 9)0.0000.0010.0050.0200.0690.1570.3190.5000.702
(8, 10)0.0000.0000.0030.0130.0480.1170.2510.4190.621
(9, 9)0.0000.0000.0030.0120.0440.1090.2380.3990.601
(9, 10)0.0000.0000.0020.0080.0290.0770.1790.3190.510
(10, 10) 0.000 0.000 0.001 0.004 0.019 0.051 0.128 0.242 0.414Reproduced from C. Eisenhart and R. Swed, Tables for Testing Randomness of Group-ing in a Sequence of Alternatives, Ann. Math. Stat., 14, 1943, by permission of theeditor.
762Appendix A Statistical Tables and Proofs
762Appendix A Statistical Tables and Proofs
Table A.18 Table for the Runs Test 763
Table A.18 (continued) P (V v when H0 is true) in the Runs Test v
(n1 , n2 )11121314151617181920
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(2, 7)
(2, 8)
(2, 9)
(2, 10)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(3, 7)
(3, 8)
(3, 9)
(3, 10)
(4, 4)
(4, 5)
(4, 6)
(4, 7)
(4, 8)
(4, 9)
(4, 10)
(5, 5)
(5, 6)1.000
(5, 7)1.000
(5, 8)1.000
(5, 9)1.000
(5, 10)1.000
(6, 6)0.9981.000
(6, 7)0.9920.9991.000
(6, 8)0.9840.9981.000
(6, 9)0.9720.9941.000
(6, 10)0.9580.9901.000
(7, 7)0.9750.9960.9991.000
(7, 8)0.9490.9880.9981.0001.000
(7, 9)0.9160.9750.9940.9991.000
(7, 10)0.8790.9570.9900.9981.000
(8, 8)0.9000.9680.9910.9991.0001.000
(8, 9)0.8430.9390.9800.9960.9991.0001.000
(8, 10)0.7820.9030.9640.9900.9981.0001.000
(9, 9)0.7620.8910.9560.9880.9971.0001.0001.000
(9, 10)0.6810.8340.9230.9740.9920.9991.0001.0001.000
(10, 10) 0.586 0.758 0.872 0.949 0.981 0.996 0.999 1.000 1.000 1.000
Table A.19 Sample Size for Two-Sided Nonparametric Tolerance Limits 1 1 0.50 0.70 0.90 0.95 0.99 0.995
0.99533648877794713251483
0.99168244388473662740
0.9534497793130146
0.90172438466472
0.85111625304247
0.8091218223134
0.7571015182427
0.706812142022
0.60469101416
0.5035781112
Reproduced from Table A25d of Wilfrid J. Dixon and Frank J. Massey, Jr., Introduction to Statistical Analysis, 3rd ed. McGraw-Hill, New York, 1969. Used with permission of McGraw-Hill Book Company.
Table A.20 Sample Size for One-Sided Nonparametric Tolerance Limits 1 1 0.50 0.70 0.95 0.99 0.995
0.9951392415989191379
0.9969120299459688
0.9514245990135
0.90712294466
0.8558192943
0.8046142131
0.753511725
0.702491320
0.602361014
0.50125710
Reproduced from Table A25e of Wilfrid J. Dixon and Frank J. Massey, Jr., Introduction to Statistical Analysis, 3rd ed. McGraw-Hill, New York, 1969. Used with permission of McGraw-Hill Book Company.
764Appendix A Statistical Tables and Proofs
764Appendix A Statistical Tables and Proofs
Table A.21 Table for Spearmans Rank Correlation Coecients 765
Table A.21 Critical Values for Spearmans Rank Correlation Coecients
n = 0.05 = 0.025 = 0.01 = 0.005
50.900
60.8290.8860.943
70.7140.7860.893
80.6430.7380.8330.881
90.6000.6830.7830.833
100.5640.6480.7450.794
110.5230.6230.7360.818
120.4970.5910.7030.780
130.4750.5660.6730.745
140.4570.5450.6460.716
150.4410.5250.6230.689
160.4250.5070.6010.666
170.4120.4900.5820.645
180.3990.4760.5640.625
190.3880.4620.5490.608
200.3770.4500.5340.591
210.3680.4380.5210.576
220.3590.4280.5080.562
230.3510.4180.4960.549
240.3430.4090.4850.537
250.3360.4000.4750.526
260.3290.3920.4650.515
270.3230.3850.4560.505
280.3170.3770.4480.496
290.3110.3700.4400.487
300.3050.3640.4320.478
Reproduced from E. G. Olds, Distribution of Sums of Squares of Rank Dierences for Small Samples, Ann. Math. Stat., 9, 1938, by permission of the editor.
Appendix A Statistical Tables and Proofs766Chart forAveragesChart for Standard DeviationsChart for RangesObs. inSampleFactors forControl LimitsFactors for Factors forCenterline Control LimitsFactors for Factors forCenterline Control LimitsnA2 A3c4 1/c4 B3 B4 B5B6d2 1/d2 d3 D3 D421.8802.6590.79791.253303.26702.6061.1280.88650.85303.26731.0231.9540.88621.128402.56802.2761.6930.59070.88802.57440.7291.6280.92131.085402.26602.0882.0590.48570.88002.28250.5771.4270.94001.063802.08901.9642.3260.42990.86402.11460.4831.2870.95151.05100.0301.9700.0291.8742.5340.39460.84802.00470.4191.1820.95941.04230.1181.8820.1131.8062.7040.36980.8330.0761.92480.3731.0990.96501.03630.1851.8150.1791.7512.8470.35120.8200.1361.86490.3371.0320.96931.03170.2391.7610.2321.7072.9700.33670.8080.1841.816100.3080.9750.97271.02810.2841.7160.2761.6693.0780.32490.7970.2231.777110.2850.9270.97541.02520.3211.6790.3131.6373.1730.31520.7870.2561.744120.2660.8860.97761.02290.3541.6460.3461.6103.2580.30690.7780.2831.717130.2490.8500.97941.02100.3821.6180.3741.5853.3360.29980.7700.3071.693140.2350.8170.98101.01940.4061.5940.3991.5633.4070.29350.7630.3281.672150.2230.7890.98231.01800.4281.5720.4211.5443.4720.28800.7560.3471.653160.2120.7630.98351.01680.4481.5520.4401.5263.5320.28310.7500.3631.637170.2030.7390.98451.01570.4661.5340.4581.5113.5880.27870.7440.3781.622180.1940.7180.98541.01480.4821.5180.4751.4963.6400.27470.7390.3911.608190.1870.6980.98621.01400.4971.5030.4901.4833.6890.27110.7340.4031.597200.1800.6800.98691.01330.5101.4900.5041.4703.7350.26770.7290.4151.585210.1730.6630.98761.01260.5231.4770.5161.4593.7780.26470.7240.4251.575220.1670.6470.98821.01190.5341.4660.5281.4483.8190.26180.7200.4341.566230.1620.6330.98871.01140.5451.4550.5391.4383.8580.25920.7160.4431.557240.1570.6190.98921.01090.5551.4450.5491.4293.8950.25670.7120.4511.548250.1530.6060.98961.01050.5651.4350.5591.4203.9310.25440.7080.4594.541Table A.22 Factors for Constructing Control Charts
Section A.24 Proof of Mean of the Hypergeometric Distribution 767
0 () Table A.23 The Incomplete Gamma Function: F (x; )= x 1 y1 ey dy
x12345678910
10.63200.26400.08000.01900.00400.00100.00000.00000.00000.0000
20.86500.59400.32300.14300.05300.01700.00500.00100.00000.0000
30.95000.80100.57700.35300.18500.08400.03400.01200.00400.0010
40.98200.90800.76200.56700.37100.21500.11100.05100.02100.0080
50.99300.96000.87500.73500.56000.38400.23800.13300.06800.0320
60.99800.98300.93800.84900.71500.55400.39400.25