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5-1 Introduction 5-2 Inference on the Means of Two Populations, Variances Known Assumptions.

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Presentation on theme: "5-1 Introduction 5-2 Inference on the Means of Two Populations, Variances Known Assumptions."— Presentation transcript:

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4 5-1 Introduction

5 5-2 Inference on the Means of Two Populations, Variances Known Assumptions

6 5-2 Inference on the Means of Two Populations, Variances Known 5-2.1 Hypothesis Testing on the Difference in Means, Variances Known

7 5-2 Inference on the Means of Two Populations, Variances Known 5-2.2 Type II Error and Choice of Sample Size

8 5-2 Inference on the Means of Two Populations, Variances Known 5-2.2 Type II Error and Choice of Sample Size

9 5-2 Inference on the Means of Two Populations, Variances Known 5-2.3 Confidence Interval on the Difference in Means, Variances Known

10 5-2 Inference on the Means of Two Populations, Variances Known

11 5-2 Inference on the Means of Two Populations, Variances Known

12 5-2 Inference on the Means of Two Populations, Variances Known 5-2.3 Confidence Interval on the Difference in Means, Variances Known Choice of Sample Size

13 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.1 Hypothesis Testing on the Difference in Means

14 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.1 Hypothesis Testing on the Difference in Means

15 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.1 Hypothesis Testing on the Difference in Means

16 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.1 Hypothesis Testing on the Difference in Means

17 5-3 Inference on the Means of Two Populations, Variances Unknown

18 5-2 Inference on the Means of Two Populations, Variances Known EX 5.4 in p227 OPTIONS NOOVP NODATE NONUMBER LS=80; DATA catalyst; input catalyst yield @@; cards; 191.50289.19 194.18290.95 192.18290.46 195.39293.21 191.79297.19 189.07297.04 194.72291.07 189.21292.75 proc ttest data=catalyst; class catalyst; var yield; title 't-test for two population means'; RUN; QUIT;

19 5-2 Inference on the Means of Two Populations, Variances Known EX 5.3 in p223 t-test for two population means The TTEST Procedure Variable: yield catalyst N Mean Std Dev Std Err Minimum Maximum 1 8 92.2550 2.3850 0.8432 89.0700 95.3900 2 8 92.7325 2.9835 1.0548 89.1900 97.1900 Diff (1-2) -0.4775 2.7009 1.3504 catalyst Method Mean 95% CL Mean Std Dev 1 92.2550 90.2611 94.2489 2.3850 2 92.7325 90.2383 95.2267 2.9835 Diff (1-2) Pooled -0.4775 -3.3739 2.4189 2.7009 Diff (1-2) Satterthwaite -0.4775 -3.3871 2.4321 catalyst Method 95% CL Std Dev 1 1.5769 4.8542 2 1.9726 6.0721 Diff (1-2) Pooled 1.9774 4.2595 Diff (1-2) Satterthwaite Method Variances DF t Value Pr > |t| Pooled Equal 14 -0.35 0.7289 Satterthwaite Unequal 13.353 -0.35 0.7292 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 7 7 1.56 0.5691

20 5-3 Inference on the Means of Two Populations, Variances Unknown

21 5-3 Inference on the Means of Two Populations, Variances Unknown

22 5-3 Inference on the Means of Two Populations, Variances Unknown

23 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 48 0 7 1 44 0 25 1 40 0 10 1 38 0 15 1 33 0 6 1 21 0 12 1 20 0 25 1 12 0 15 1 1 0 7 1 18 PROC TTEST DATA=ARSENIC; CLASS AREA; VAR ARSENIC; TITLE 'EXAMPLE 5-5'; RUN; QUIT;

24 5-3 Inference on the Means of Two Populations, Variances Unknown 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

25 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.1 Hypothesis Testing on the Difference in Means

26 5-3 Inference on the Means of Two Populations, Variances Unknown

27 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.2 Type II Error and Choice of Sample Size

28 Standardized Difference, d (a) OC Curves for a Two—Sided t—Test (α = 0.05 ) Chart VOperating Characteristic Curves for the t-Test

29 Standardized Difference, d (b) OC Curves for a Two-Sided t—Test (α = 0.01)

30 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.3 Confidence Interval on the Difference in Means

31 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.3 Confidence Interval on the Difference in Means

32 5-3 Inference on the Means of Two Populations, Variances Unknown 5-3.3 Confidence Interval on the Difference in Means

33 5-3 Inference on the Means of Two Populations, Variances Unknown OPTIONS NOOVP NODATE NONUMBER LS=80; DATA EX520; INPUT TYPE TEMP @@; CARDS; 12062177 11882197 12052206 11872201 11942180 11932176 12072185 11852200 11892197 12132192 11922198 12102188 11942189 11782203 12052192 PROC SORT; BY TYPE; PROC UNIVARIATE NORMAL; VAR TEMP; BY TYPE; TITLE 'NORMALITY CHECK'; PROC TTEST DATA=EX520 SIDES=U; CLASSTYPE; VARTEMP; TITLE 'EXERCISE 520'; RUN; QUIT; EX 5-20 (P235)

34 5-3 Inference on the Means of Two Populations, Variances Unknown 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

35 5-3 Inference on the Means of Two Populations, Variances Unknown 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

36 5-4 The Paired t-Test

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39 OPTIONS NOOVP NODATE NONUMBER LS=80; DATA STRENGTH; INPUT K L @@; DIFF = K-L; CARDS; 1.1861.061 1.1510.992 1.3221.063 1.3391.062 1.21.065 1.4021.178 1.3651.037 1.5371.086 1.5591.052 PROC 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;

40 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

41 5-4 The Paired t-Test

42 Paired Versus Unpaired Comparisons

43 5-4 The Paired t-Test Confidence Interval for  D

44 5-4 The Paired t-Test

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46 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 (X 1j, X 2j ), 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.

47 5-5 Inference on the Ratio of Variances of Two Normal Populations 5-5.1 The F Distribution We wish to test the hypotheses: The development of a test procedure for these hypotheses requires a new probability distribution, the F distribution.

48 5-5 Inference on the Ratio of Variances of Two Normal Populations 5-5.1 The F Distribution

49 5-5 Inference on the Ratio of Variances of Two Normal Populations 5-5.1 The F Distribution

50 5-5 Inference on the Ratio of Variances of Two Normal Populations The Test Procedure

51 5-5 Inference on the Ratio of Variances of Two Normal Populations The Test Procedure

52 5-5 Inference on the Ratio of Variances of Two Normal Populations The Test Procedure

53 5-5 Inference on the Ratio of Variances of Two Normal Populations 5-5.2 Confidence Interval on the Ratio of Two Variances

54 5-5 Inference on the Ratio of Variances of Two Normal Populations

55 5-5 Inference on the Ratio of Variances of Two Normal Populations

56 5-6 Inference on Two Population Proportions 5-6.1 Hypothesis Testing on the Equality of Two Binomial Proportions

57 5-6 Inference on Two Population Proportions 5-6.1 Hypothesis Testing on the Equality of Two Binomial Proportions

58 5-6 Inference on Two Population Proportions

59 5-6 Inference on Two Population Proportions

60 5-6 Inference on Two Population Proportions

61 5-6 Inference on Two Population Proportions 5-6.2 Type II Error and Choice of Sample Size

62 5-6 Inference on Two Population Proportions 5-6.2 Type II Error and Choice of Sample Size

63 5-6 Inference on Two Population Proportions 5-6.2 Type II Error and Choice of Sample Size

64 5-6 Inference on Two Population Proportions 5-6.3 Confidence Interval on the Difference in Binomial Proportions

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