1. 1 1 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2. 2 2 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 10, Part A Statistical Inference About Means and Proportions With Two Populations n Inferences About the Difference Between Two Population Means:  1 and  2 Known Two Population Means:  1 and  2 Known n Inferences About the Difference Between Two Population Means:  1 and  2 Unknown Two Population Means:  1 and  2 Unknown

3. 3 3 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Estimating the Difference Between Two Population Means Let  1 equal the mean of population 1 and  2 equal Let  1 equal the mean of population 1 and  2 equal the mean of population 2. the mean of population 2. n The difference between the two population means is  1 -  2.  1 -  2. To estimate  1 -  2, we will select a simple random To estimate  1 -  2, we will select a simple random sample of size n 1 from population 1 and a simple sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. random sample of size n 2 from population 2. n Let equal the mean of sample 1 and equal the mean of sample 2. mean of sample 2. The point estimator of the difference between the The point estimator of the difference between the means of the populations 1 and 2 is. means of the populations 1 and 2 is.

4. 4 4 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Expected Value Sampling Distribution of n Standard Deviation (Standard Error) where:  1 = standard deviation of population 1  2 = standard deviation of population 2  2 = standard deviation of population 2 n 1 = sample size from population 1 n 1 = sample size from population 1 n 2 = sample size from population 2 n 2 = sample size from population 2

5. 5 5 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Interval Estimate Interval Estimation of  1 -  2 :  1 and  2 Known where: 1 -  is the confidence coefficient 1 -  is the confidence coefficient

6. 6 6 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Par, Inc. Interval Estimation of  1 -  2 :  1 and  2 Known In a test of driving distance using a mechanical In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide. Par, Inc. is a manufacturer of golf equipment and Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.”

7. 7 7 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Par, Inc. Interval Estimation of  1 -  2 :  1 and  2 Known Sample Size Sample Mean Sample #1 Par, Inc. Sample #2 Rap, Ltd. 120 balls 80 balls 120 balls 80 balls 275 yards 258 yards Based on data from previous driving distance Based on data from previous driving distance tests, the two population standard deviations are known with  1 = 15 yards and  2 = 20 yards.

8. 8 8 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1 -  2 :  1 and  2 Known n Example: Par, Inc. Let us develop a 95% confidence interval estimate Let us develop a 95% confidence interval estimate of the difference between the mean driving distances of the two brands of golf ball.

9. 9 9 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Estimating the Difference Between Two Population Means  1 –  2 = difference between the mean distances the mean distances x 1 - x 2 = Point Estimate of  1 –  2 Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls for the Par golf balls Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls for the Par golf balls

10. 10 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Point Estimate of  1 -  2 Point estimate of  1   2 = where:  1 = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls = 275  258 = 17 yards

11. 11 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1 -  2 :   1 and   2 Known We are 95% confident that the difference between We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls is 11.86 to 22.14 yards. 17 + 5.14 or 11.86 yards to 22.14 yards

12. 12 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Formula Worksheet Interval Estimation of  1 -  2 :   1 and   2 Known Note: Rows 16-121 are not shown. ABCDE 1ParRapPar, Inc.Rap, Ltd. 2255266Sample Size=COUNT(A2:A121)=COUNT(B2:B81) 3270238Sample Mean=AVERAGE(A2:A121)=AVERAGE(B2:B81) 4294243 5245277Popul. Std. Dev.1520 6300275Standard Error=SQRT(D5^2/D2+E5^2/E2) 7262244 8281239Confid. Coeff.0.95 9257242Level of Signif.=1-D8 10268280z Value=NORM.S.INV(1-D9/2) 11295261Margin of Error=D10*D6 12249276 13291241Pt. Est. of Diff.=D3-E3 14289273Lower Limit=D13-D11 15282248Upper Limit=D13+D11

13. 13 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Value Worksheet Interval Estimation of  1 -  2 :   1 and   2 Known Note: Rows 16-121 are not shown.

14. 14 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Known Hypotheses Hypotheses Left-tailedRight-tailedTwo-tailed Test Statistic Test Statistic

15. 15 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Par, Inc. Hypothesis Tests About  1   2 :  1 and  2 Known Can we conclude, using  =.01, that the mean Can we conclude, using  =.01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. Golf balls?

16. 16 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. H 0 :  1 -  2 < 0  H a :  1 -  2 > 0 where:  1 = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls 1. Develop the hypotheses. p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About  1   2 :  1 and  2 Known 2. Specify the level of significance.  =.01 Right-tailedtestRight-tailedtest

17. 17 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 3. Compute the value of the test statistic. Hypothesis Tests About  1   2 :  1 and  2 Known p –Value and Critical Value Approaches p –Value and Critical Value Approaches

18. 18 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. p –Value Approach p –Value Approach 4. Compute the p –value. For z = 6.49, the p –value <.0001. Hypothesis Tests About  1   2 :  1 and  2 Known 5. Determine whether to reject H 0. Because p –value <  =.01, we reject H 0. At the.01 level of significance, the sample evidence At the.01 level of significance, the sample evidence indicates the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls.

19. 19 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Known 5. Determine whether to reject H 0. Because z = 6.49 > 2.33, we reject H 0. Critical Value Approach Critical Value Approach For  =.01, z.01 = 2.33 4. Determine the critical value and rejection rule. Reject H 0 if z > 2.33 The sample evidence indicates the mean driving The sample evidence indicates the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls.

20. 20 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose z -Test: Two Sample for Means from the list of Analysis Tools from the list of Analysis Tools Step 4 When the z -Test: Two Sample for Means dialog box appears: dialog box appears: (see details on next slide) (see details on next slide) Excel’s “ z -Test: Two Sample for Means” Tool

21. 21 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Excel’s “ z -Test: Two Sample for Means” Tool n Excel Dialog Box

22. 22 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Value Worksheet Note: Rows 14-121 are not shown. Excel’s “ z -Test: Two Sample for Means” Tool

23. 23 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1 -  2 :  1 and  2 Unknown When  1 and  2 are unknown, we will: replace z  /2 with t  /2. replace z  /2 with t  /2. use the sample standard deviations s 1 and s 2 use the sample standard deviations s 1 and s 2 as estimates of  1 and  2, and

24. 24 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Where the degrees of freedom for t  /2 are: Interval Estimation of  1 -  2 :  1 and  2 Unknown n Interval Estimate

25. 25 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Difference Between Two Population Means:  1 and  2 Unknown Specific Motors of Detroit has developed a new Specific Motors of Detroit has developed a new automobile known as the M car. 24 M cars and 28 J cars (from Japan) were road tested to compare miles-per-gallon (mpg) performance. The sample statistics are shown on the next slide. n Example: Specific Motors

26. 26 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Difference Between Two Population Means:  1 and  2 Unknown n Example: Specific Motors Sample Size Sample Mean Sample Std. Dev. Sample #1 M Cars Sample #2 J Cars 24 cars 2 8 cars 24 cars 2 8 cars 29.8 mpg 27.3 mpg 2.56 mpg 1.81 mpg

27. 27 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Difference Between Two Population Means:  1 and  2 Unknown Let us develop a 90% confidence interval estimate Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile. n Example: Specific Motors

28. 28 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Point estimate of  1   2 = Point Estimate of  1   2 where:  1 = mean miles-per-gallon for the population of M cars population of M cars  2 = mean miles-per-gallon for the population of J cars population of J cars = 29.8 - 27.3 = 2.5 mpg

29. 29 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1   2 :  1 and  2 Unknown The degrees of freedom for t  /2 are: With  /2 =.05 and df = 24, t  /2 = 1.711

30. 30 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Interval Estimation of  1   2 :  1 and  2 Unknown We are 90% confident that the difference between We are 90% confident that the difference between the miles-per-gallon performances of M cars and J cars is 1.431 to 3.569 mpg. 2.5 + 1.069 or 1.431 to 3.569 mpg

31. 31 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Formula Worksheet Interval Estimation of  1   2 :  1 and  2 Unknown ABCDE 1MJPar, Inc.Rap, Ltd. 226.125.6Sample Size=COUNT(A2:A25)=COUNT(B2:B29) 332.528.1Sample Mean=AVERAGE(A2:A25)=AVERAGE(B2:B29) 431.827.9Sample Std. Dev.=STDEV(A2:A25)=STDEV(B2:B29) 527.625.3 628.530.1Est. of Variance=D4^2/D2+E4^2/E2 733.627.5Standard Error=SQRT(D6) 831.726.0 925.228.8Confid. Coeff.0.90 1026.030.6Level of Signif.=1-D9 1132.024.4Degr. of Freedom 1231.727.3t Value=T.INV.2T(D10,D11) 1330.427.5Margin of Error=D12*D7 1427.626.3 1532.325.5Point Est. of Diff.=D3-E3 1630.626.3Lower Limit=D15-D13 1729.524.3Upper Limit=D15+D13 =D6^2/((1/(D2-1))*(D4^2/D2)^2+(1/(E2-1))*(E4^2/E2)^2)) Note: Rows 18-29 are not shown.

32. 32 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Formula Worksheet Interval Estimation of  1   2 :  1 and  2 Unknown Note: Rows 18-29 are not shown.

33. 33 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Unknown n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

34. 34 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Example: Specific Motors Hypothesis Tests About  1   2 :  1 and  2 Unknown Can we conclude, using a.05 level of significance, Can we conclude, using a.05 level of significance, that the miles-per-gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars?

35. 35 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. H 0 :  1 -  2 < 0  H a :  1 -  2 > 0 where:  1 = mean mpg for the population of M cars  2 = mean mpg for the population of J cars 1. Develop the hypotheses. p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About  1   2 :  1 and  2 Unknown Right-tailedtestRight-tailedtest

36. 36 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05 p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About  1   2 :  1 and  2 Unknown

37. 37 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About  1   2 :  1 and  2 Unknown p –Value Approach p –Value Approach 4. Compute the p –value. The degrees of freedom for t  are: Because t = 4.003 > t.005 = 1.683, the p –value t.005 = 1.683, the p –value <.005.

38. 38 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 5. Determine whether to reject H 0. We are at least 95% confident that the miles-per- gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars. We are at least 95% confident that the miles-per- gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars. p –Value Approach p –Value Approach Because p –value <  =.05, we reject H 0. Hypothesis Tests About  1   2 :  1 and  2 Unknown

39. 39 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 4. Determine the critical value and rejection rule. Critical Value Approach Critical Value Approach Hypothesis Tests About  1   2 :  1 and  2 Unknown For  =.05 and df = 41, t.05 = 1.683 Reject H 0 if t > 1.683 5. Determine whether to reject H 0. Because 4.003 > 1.683, we reject H 0. We are at least 95% confident that the miles-per- gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars. We are at least 95% confident that the miles-per- gallon ( mpg ) performance of M cars is greater than the miles-per-gallon performance of J cars.

40. 40 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose t -Test: Two-Sample Assuming Unequal Variances from the list of Unequal Variances from the list of Analysis Tools Analysis Tools Excel’s “ z -Test: Two-Sample Assuming Unequal Variances” Tool Step 4 When the t -Test: Two-Sample Assuming Unequal Variances dialog box appears: Unequal Variances dialog box appears: (see details on next slide) (see details on next slide)

41. 41 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Dialog Box Excel’s “ z -Test: Two-Sample Assuming Unequal Variances” Tool

42. 42 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Excel Value Worksheet Note: Rows 14-121 are not shown. Excel’s “ z -Test: Two-Sample Assuming Unequal Variances” Tool

43. 43 Slide © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 10, Part A