5-3 Inference on the Means of Two Populations, Variances Unknown.

5-3 Inference on the Means of Two Populations, Variances Unknown


OPTIONS NOOVP NODATE NONUMBER LS=80;PROC FORMAT;

VALUE MR 0='PHX' 1='RuralAZ';DATA ARSENIC;INPUT AREA ARSENIC @@;FORMAT AREA MR.;CARDS;0 3 1 480 7 1 440 25 1 400 10 1 380 15 1 330 6 1 210 12 1 200 25 1 120 15 1 10 7 1 18PROC TTEST DATA=ARSENIC;CLASS AREA;VAR ARSENIC;TITLE 'EXAMPLE 5-5';RUN; QUIT;


EXAMPLE 5-5 The TTEST Procedure Variable: ARSENIC

AREA N Mean Std Dev Std Err Minimum Maximum PHX 10 12.5000 7.6340 2.4141 3.0000 25.0000 RuralAZ 10 27.5000 15.3496 4.8540 1.0000 48.0000 Diff (1-2) -15.0000 12.1221 5.4212

AREA Method Mean 95% CL Mean Std Dev PHX 12.5000 7.0390 17.9610 7.6340 RuralAZ 27.5000 16.5195 38.4805 15.3496 Diff (1-2) Pooled -15.0000 -26.3894 -3.6106 12.1221 Diff (1-2) Satterthwaite -15.0000 -26.6941 -3.3059

AREA Method 95% CL Std Dev PHX 5.2509 13.9367 RuralAZ 10.5580 28.0224 Diff (1-2) Pooled 9.1596 17.9264 Diff (1-2) Satterthwaite

Method Variances DF t Value Pr > |t| Pooled Equal 18 -2.77 0.0127 Satterthwaite Unequal 13.196 -2.77 0.0158

Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 9 9 4.04 0.0494


5-3.1 Hypothesis Testing on the Difference in Means

5-3 Inference on the Means of Two Populations, Variances Unknown5-3.2 Type II Error and Choice of Sample Size

Standardized Difference, d

(a) OC Curves for a Two—Sided t—Test (α = 0.05 )

Chart V Operating Characteristic Curves for the t-Test

Standardized Difference, d

(b) OC Curves for a Two-Sided t—Test (α = 0.01)

http://www.fhwa.dot.gov/publications/research/infrastructure/pavements/pccp/02095/captions.cfm#figg3

5-3 Inference on the Means of Two Populations, Variances Unknown5-3.3 Confidence Interval on the Difference in Means


OPTIONS NOOVP NODATE NONUMBER LS=80;DATA EX520;INPUT TYPE TEMP @@;CARDS;1 206 2 177 1 188 2 197 1 205 2 206 1 187 2 2011 194 2 180 1 193 2 176 1 207 2 185 1 185 2 2001 189 2 197 1 213 2 192 1 192 2 198 1 210 2 1881 194 2 189 1 178 2 203 1 205 2 192 PROC SORT; BY TYPE;PROC UNIVARIATE NORMAL PLOT; VAR TEMP; BY TYPE;TITLE 'NORMALITY CHECK';PROC TTEST DATA=EX520 SIDES=U; CLASS TYPE; VAR TEMP;TITLE 'EXERCISE 520';RUN; QUIT;

EX 5-20 (P235)


NORMALITY CHECK

------------------------------------ TYPE=1 ------------------------------------

UNIVARIATE 프로시저 변수 : TEMP

적률

N 15 가중합 15 평균 196.4 관측치 합 2946 표준편차 10.4799128 분산 109.828571 왜도 0.05341203 첨도 -1.126598 제곱합 580132 수정 제곱합 1537.6 변동계수 5.33600446 평균의 표준오차 2.70590184

정규성 검정

검정 ---- 통계량 ---- -------p-값 -------

Shapiro-Wilk W 0.939894 Pr < W 0.3810 Kolmogorov-Smirnov D 0.194068 Pr > D 0.1304 Cramer-von Mises W-Sq 0.087134 Pr > W-Sq 0.1557 Anderson-Darling A-Sq 0.463122 Pr > A-Sq 0.2270

NORMALITY CHECK

------------------------------------ TYPE=2 ------------------------------------

UNIVARIATE 프로시저 변수 : TEMP

적률

N 15 가중합 15 평균 192.066667 관측치 합 2881 표준편차 9.4375138 분산 89.0666667 왜도 -0.4020429 첨도 -0.9023837 제곱합 554591 수정 제곱합 1246.93333 변동계수 4.91366564 평균의 표준오차 2.43675558

정규성 검정

검정 ---- 통계량 ---- -------p- 값 -------

Shapiro-Wilk W 0.947736 Pr < W 0.4895 Kolmogorov-Smirnov D 0.166088 Pr > D >0.1500 Cramer-von Mises W-Sq 0.043562 Pr > W-Sq >0.2500 Anderson-Darling A-Sq 0.295176 Pr > A-Sq >0.2500


줄기 잎 # 상자그림 21 03 2 | 20 5567 4 +-----+ 20 | | 19 | + | 19 2344 4 *-----* 18 5789 4 +-----+ 18 | 17 8 1 | ----+----+----+----+ 값 : ( 줄기 . 잎 )*10**+1

정규 확률도 212.5+ *++++* | * * * *+++ | +++++ | +++++ | +++** * * | * +*+* * | +++++ 177.5+ +++*+ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2

줄기 잎 # 상자그림 20 6 1 | 20 013 3 +-----+ 19 778 3 | | 19 22 2 *--+--* 18 589 3 +-----+ 18 0 1 | 17 67 2 | ----+----+----+----+ 값 : ( 줄기 . 잎 )*10**+1

정규 확률도 207.5+ +++*++ | * *++*+ | **+*+++ 192.5+ *+*+++ | *+*+* | +++*+ 177.5+ +*++++* +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2

------------------------------------ TYPE=1 ------------------------------------ ------------------------------------ TYPE=2 ------------------------------------


Variable: TEMP

TYPE N Mean Std Dev Std Err Minimum Maximum 1 15 196.4 10.4799 2.7059 178.0 213.0 2 15 192.1 9.4375 2.4368 176.0 206.0 Diff (1-2) 4.3333 9.9723 3.6414

TYPE Method Mean 95% CL Mean Std Dev 1 196.4 190.6 202.2 10.4799 2 192.1 186.8 197.3 9.4375 Diff (1-2) Pooled 4.3333 -1.8611 Infty 9.9723 Diff (1-2) Satterthwaite 4.3333 -1.8634 Infty

TYPE Method 95% CL Std Dev 1 7.6726 16.5279 2 6.9095 14.8839 Diff (1-2) Pooled 7.9138 13.4871 Diff (1-2) Satterthwaite

Method Variances DF t Value Pr > t Pooled Equal 28 1.19 0.1220 Satterthwaite Unequal 27.698 1.19 0.1221

Equality of Variances

Method Num DF Den DF F Value Pr > F Folded F 14 14 1.23 0.7004

or

Inference on Two Population

H0 : m1 = m2

Both s’s Known

Both n’s Large

Z –TestNormal Distribution

Use S for s If s unknown

t –TestPooled Variance

Wilcoxon-Mann-Whitney Test

t –TestSatterthwaite

s1 = s2

F Test

Both X’s Normal

YES

YES

YES

YES

NO

NO

NO

NO


Sample Problem

The number of visitors to Carlsbad Caverns were counted for a one-week period that included the forth of July in 2009 and in 2010. Treat these data as random samples and use the Wilcoxon-Mann-Whitney rank sum test to see if the mean number of visitors is the same for both years. Use and state the p-value.

Visitors;Week of

July 4, 2009

Visitors;Week of

July 4, 2010

397286268254571604384

314257278252613646253


1. H0: m1 = m2

H1: m1 m2

2. 3. Test Statistic where =

=

=0.5572 = 18.4761

4. Decision Rule: If |T|>T1-a, n1+n2-2, then Reject H0

T0.95, 12 = 1.7823

5. Conclusion: Since |T|=0.5572< T0.95, 12 = 1.7823, fail to reject H0.

X1 R1 X2 R2

397286268254571604384

10753

11129

314257278252613646253

8461

13142

= 8.14 = 6.86

SR12 = 10.8095, SR1 = 3.29

SR22 = 26.1428, SR2 = 5.11


DATA CARLSBAD; INPUT YEAR COUNT @@; CARDS;2009 397 2009 286 2009 268 2009 254 2009 571 2009 604 2009 3842010 314 2010 257 2010 278 2010 252 2010 613 2010 646 2010 253PROC UNIVARIATE DATA=CARLSBAD NORMAL; VAR COUNT; BY YEAR;TITLE 'PROBLEM ASSUMING NORMALITY';PROC TTEST DATA=CARLSBAD; CLASS YEAR; VAR COUNT;PROC RANK DATA=CARLSBAD OUT=RANKED; VAR COUNT;PROC TTEST DATA=RANKED; CLASS YEAR; VAR COUNT;TITLE 'Problem using Wilcoxon-Mann-Whitney test';RUN; QUIT;


PROBLEM ASSUMING NORMALITY ------------------------------------------ YEAR=2009 ------------------------------------------ UNIVARIATE 프로시저 변수 : COUNT 적률 N 7 가중합 7 평균 394.857143 관측치 합 2764 표준편차 142.987678 분산 20445.4762 왜도 0.67728241 첨도 -1.3040573 제곱합 1214058 수정 제곱합 122672.857 변동계수 36.212509 평균의 표준오차 54.0442625

정규성 검정 검정 ---- 통계량 ---- -------p- 값 ------- Shapiro-Wilk W 0.864041 Pr < W 0.1645 Kolmogorov-Smirnov D 0.208307 Pr > D >0.1500 Cramer-von Mises W-Sq 0.069546 Pr > W-Sq 0.2470 Anderson-Darling A-Sq 0.44369 Pr > A-Sq 0.2043 ------------------------------------------ YEAR=2010 ------------------------------------------ UNIVARIATE 프로시저 변수 : COUNT 적률 N 7 가중합 7 평균 373.285714 관측치 합 2613 표준편차 176.602864 분산 31188.5714 왜도 1.18136027 첨도 -0.8247496 제곱합 1162527 수정 제곱합 187131.429 변동계수 47.3103729 평균의 표준오차 66.7496083 정규성 검정 검정 ---- 통계량 ---- -------p- 값 ------- Shapiro-Wilk W 0.70274 Pr < W 0.0040 Kolmogorov-Smirnov D 0.345737 Pr > D 0.0124 Cramer-von Mises W-Sq 0.187549 Pr > W-Sq 0.0050 Anderson-Darling A-Sq 1.012182 Pr > A-Sq <0.0050


The TTEST Procedure Variable: COUNT YEAR N Mean Std Dev Std Err Minimum Maximum 2009 7 394.9 143.0 54.0443 254.0 604.0 2010 7 373.3 176.6 66.7496 252.0 646.0 Diff (1-2) 21.5714 160.7 85.8853 YEAR Method Mean 95% CL Mean Std Dev 95% CL Std Dev 2009 394.9 262.6 527.1 143.0 92.1403 314.9 2010 373.3 210.0 536.6 176.6 113.8 388.9 Diff (1-2) Pooled 21.5714 -165.6 208.7 160.7 115.2 265.2 Diff (1-2) Satterthwaite 21.5714 -166.5 209.6 Method Variances DF t Value Pr > |t| Pooled Equal 12 0.25 0.8059 Satterthwaite Unequal 11.502 0.25 0.8061 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 6 6 1.53 0.6210 _____________________________________________________________________________________

Problem using Wilcoxon-Mann-Whitney test The TTEST Procedure Variable: COUNT (Values of COUNT Were Replaced by Ranks) YEAR N Mean Std Dev Std Err Minimum Maximum 2009 7 8.1429 3.2878 1.2427 3.0000 12.0000 2010 7 6.8571 5.1130 1.9325 1.0000 14.0000 Diff (1-2) 1.2857 4.2984 2.2976 YEAR Method Mean 95% CL Mean Std Dev 95% CL Std Dev 2009 8.1429 5.1022 11.1836 3.2878 2.1186 7.2399 2010 6.8571 2.1284 11.5859 5.1130 3.2948 11.2592 Diff (1-2) Pooled 1.2857 -3.7203 6.2917 4.2984 3.0823 7.0955 Diff (1-2) Satterthwaite 1.2857 -3.8176 6.3890 Method Variances DF t Value Pr > |t| Pooled Equal 12 0.56 0.5861 Satterthwaite Unequal 10.237 0.56 0.5878 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 6 6 2.42 0.3067

5-4 The Paired t-Test

• A special case of the two-sample t-tests of Section 5-3 occurs when the observations on the two populations of interest are collected in pairs.

• Each pair of observations, say (X1j , X2j ), is taken under homogeneous conditions, but these conditions may change from one pair to another.

• The test procedure consists of analyzing the differences between hardness readings on each specimen.


OPTIONS NOOVP NODATE NONUMBER LS=80;DATA STRENGTH;INPUT K L @@; DIFF = K-L;CARDS;1.186 1.0611.151 0.9921.322 1.0631.339 1.0621.2 1.0651.402 1.1781.365 1.0371.537 1.0861.559 1.052PROC UNIVARIATE DATA=STRENGTH NORMAL; VAR DIFF;TITLE 'PAIRED T-TEST BY PROC UNIVARIATE';PROC TTEST DATA=STRENGTH; PAIRED K*L;TITLE 'PAIRED TTEST BY PROC TTEST';RUN; QUIT;

5-4 The Paired t-Test PAIRED T-TEST BY PROC UNIVARIATE UNIVARIATE 프로시저 변수 : DIFF

적률 N 9 가중합 9 평균 0.27388889 관측치 합 2.465 표준편차 0.13509945 분산 0.01825186 왜도 0.70116761 첨도 -0.5595974 제곱합 0.821151 수정 제곱합 0.14601489 변동계수 49.3263708 평균의 표준오차 0.04503315

위치모수 검정 : Mu0=0

검정 -- 통계량 --- -------p- 값 ------- 스튜던트의 t t 6.081939 Pr > |t| 0.0003

정규성 검정

검정 ---- 통계량 ---- -------p- 값 ------- Shapiro-Wilk W 0.916781 Pr < W 0.3663 Kolmogorov-Smirnov D 0.157481 Pr > D >0.1500 --------------------------------------------------------------------------------------------------- PAIRED TTEST BY PROC TTEST The TTEST Procedure Difference: K - L

N Mean Std Dev Std Err Minimum Maximum 9 0.2739 0.1351 0.0450 0.1250 0.5070

Mean 95% CL Mean Std Dev 95% CL Std Dev 0.2739 0.1700 0.3777 0.1351 0.0913 0.2588

DF t Value Pr > |t| 8 6.08 0.0003


Paired Versus Unpaired Comparisons


Confidence Interval for D


First Second D

165156165135134131130126120120118115109

139132134133130133130125122119114116105

26243124-201-214-13

Sample Example:

An insurance adjuster wants to compare estimates from two different repair garages for minor repairs on automobiles. Thirteen pairs of estimated are available.(a) State the appropriate null and alternative hypothesis to see

if there is any difference in the mean estimated of the two garages. Let a =0.05 and test the null hypothesis with the Wilcoxon signed ranks test. State the p-value.

(b) Check the differences in estimates from the two garages for normality.

(c) Based on the results of part (b), the paired t test should not be applied to these data: however, compute the paired t test to test the null hypothesis on part (a) and compare it with the results of the Wilcoxon signed ranks test.

SD = 11.6619


1. H0: mD = 0 H1: mD ≠ 0

2.

3. Test Statistic (Wilcoxon Signed Ranks Test) where

4. Decision Rule:

Reject if |T|>Ta/2, n-1. Here, t0.025, 12 = 2.178.

5. Conclusion

= = 2.55

Since T=2.55> t0.025, 12 = 2.178, reject H0.

First Second D |D| |R| R

165156165135134131130126120120118115109

139132134133130133130125122119114116105

26243124-201-214-13

2624312420121413

1211136

9.561363

9.538

1211136

9.5- 613

- 63

9.5- 38

∑𝑖=1

𝑛

𝑅𝑖=61 = 4.69

∑𝑖=1

𝑛

𝑅𝑖2=2269SR= 6.63


OPTIONS NOOVP NODATE NONUMBER LS=80;DATA INSURE; INPUT FIRST SECOND @@; DIFF=FIRST-SECOND; IF DIFF<0 THEN IND=1; ELSE IND=0; ABSDIFF=ABS(DIFF); CARDS;165 139 156 132 165 134 135 133 134 130 131 133 130 130126 125 120 122 120 119 118 114 115 116 108 105PROC UNIVARIATE DATA=INSURE NORMAL; VAR DIFF;TITLE 'normality check and t-test';PROC RANK DATA=INSURE OUT=RINSURE; VAR ABSDIFF;DATA RINSURE; SET RINSURE; IF IND=1 THEN ABSDIFF=-ABSDIFF;PROC UNIVARIATE DATA=RINSURE; VAR ABSDIFF;TITLE 'Wilcoxon Signed Ranks Test';RUN; QUIT;


normality check and t-test UNIVARIATE 프로시저 변수 : DIFF 적률 N 13 가중합 13 평균 7 관측치 합 91 표준편차 11.6619038 분산 136 왜도 1.40385807 첨도 0.31339454 제곱합 2269 수정 제곱합 1632 변동계수 166.598626 평균의 표준오차 3.23443016


검정 -- 통계량 --- -------p- 값 -------

스튜던트의 t t 2.164214 Pr > |t| 0.0513 부호 M 3 Pr >= |M| 0.1460 부호 순위 S 27 Pr >= |S| 0.0332

정규성 검정

검정 ---- 통계량 ---- -------p- 값 -------

Shapiro-Wilk W 0.714134 Pr < W 0.0008 Kolmogorov-Smirnov D 0.370737 Pr > D <0.0100 Cramer-von Mises W-Sq 0.335966 Pr > W-Sq <0.0050 Anderson-Darling A-Sq 1.740466 Pr > A-Sq <0.0050


Wilcoxon Signed Ranks Test

UNIVARIATE 프로시저 변수 : ABSDIFF (Values of ABSDIFF Were Replaced by Ranks)

적률

N 13 가중합 13 평균 4.69230769 관측치 합 61 표준편차 6.63494053 분산 44.0224359 왜도 -0.50062 첨도 -1.0648238 제곱합 814.5 수정 제곱합 528.269231 변동계수 141.400372 평균의 표준오차 1.84020141


검정 -- 통계량 --- -------p- 값 -------

스튜던트의 t t 2.549888 Pr > |t| 0.0255 부호 M 3.5 Pr >= |M| 0.0923 부호 순위 S 30.5 Pr >= |S| 0.0310

5-3 Inference on the Means of Two Populations, Variances Unknown.

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Transcript of 5-3 Inference on the Means of Two Populations, Variances Unknown.