1. 1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2009 Thomson South-Western. All Rights Reserved Chapter 10, Part A Comparisons Involving Means, Experimental Design and Analysis of Variance 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: Matched Samples Two Population Means: Matched Samples n Inferences About the Difference Between Two Population Means:  1 and  2 Unknown Two Population Means:  1 and  2 Unknown

3. 3 3 Slide © 2009 Thomson South-Western. All Rights Reserved Inferences About the Difference Between Two Population Means:  1 and  2 Known Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

4. 4 4 Slide © 2009 Thomson South-Western. All Rights Reserved 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.

5. 5 5 Slide © 2009 Thomson South-Western. All Rights Reserved 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

6. 6 6 Slide © 2009 Thomson South-Western. All Rights Reserved n Interval Estimate Interval Estimation of  1 -  2 :  1 and  2 Known where: 1 -  is the confidence coefficient 1 -  is the confidence coefficient

7. 7 7 Slide © 2009 Thomson South-Western. All Rights Reserved 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.” n Example: Par, Inc.

8. 8 8 Slide © 2009 Thomson South-Western. All Rights Reserved 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.

9. 9 9 Slide © 2009 Thomson South-Western. All Rights Reserved 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.

10. 10 Slide © 2009 Thomson South-Western. All Rights Reserved 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

11. 11 Slide © 2009 Thomson South-Western. All Rights Reserved 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

12. 12 Slide © 2009 Thomson South-Western. All Rights Reserved 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

13. 13 Slide © 2009 Thomson South-Western. All Rights Reserved Hypothesis Tests About  1   2 :  1 and  2 Known Hypotheses Hypotheses Left-tailedRight-tailedTwo-tailed Test Statistic Test Statistic

14. 14 Slide © 2009 Thomson South-Western. All Rights Reserved 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?

15. 15 Slide © 2009 Thomson South-Western. All Rights Reserved 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

16. 16 Slide © 2009 Thomson South-Western. All Rights Reserved 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

17. 17 Slide © 2009 Thomson South-Western. All Rights Reserved 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.

18. 18 Slide © 2009 Thomson South-Western. All Rights Reserved 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.

19. 19 Slide © 2009 Thomson South-Western. All Rights Reserved Inferences About the Difference Between Two Population Means:  1 and  2 Unknown Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

20. 20 Slide © 2009 Thomson South-Western. All Rights Reserved 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

21. 21 Slide © 2009 Thomson South-Western. All Rights Reserved Where the degrees of freedom for t  /2 are: Interval Estimation of  1 -  2 :  1 and  2 Unknown n Interval Estimate

22. 22 Slide © 2009 Thomson South-Western. All Rights Reserved n Example: Specific Motors 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 ar shown on the next slide.

23. 23 Slide © 2009 Thomson South-Western. All Rights Reserved 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

24. 24 Slide © 2009 Thomson South-Western. All Rights Reserved 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

25. 25 Slide © 2009 Thomson South-Western. All Rights Reserved 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

26. 26 Slide © 2009 Thomson South-Western. All Rights Reserved 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

27. 27 Slide © 2009 Thomson South-Western. All Rights Reserved 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

28. 28 Slide © 2009 Thomson South-Western. All Rights Reserved Hypothesis Tests About  1   2 :  1 and  2 Unknown n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

29. 29 Slide © 2009 Thomson South-Western. All Rights Reserved 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?

30. 30 Slide © 2009 Thomson South-Western. All Rights Reserved 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

31. 31 Slide © 2009 Thomson South-Western. All Rights Reserved 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

32. 32 Slide © 2009 Thomson South-Western. All Rights Reserved 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.

33. 33 Slide © 2009 Thomson South-Western. All Rights Reserved 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

34. 34 Slide © 2009 Thomson South-Western. All Rights Reserved 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?.

35. 35 Slide © 2009 Thomson South-Western. All Rights Reserved With a matched-sample design each sampled item With a matched-sample design each sampled item provides a pair of data values. provides a pair of data values. This design often leads to a smaller sampling error This design often leads to a smaller sampling error than the independent-sample design because than the independent-sample design because variation between sampled items is eliminated as a variation between sampled items is eliminated as a source of sampling error. source of sampling error. Inferences About the Difference Between Two Population Means: Matched Samples

36. 36 Slide © 2009 Thomson South-Western. All Rights Reserved n Example: Express Deliveries Inferences About the Difference Between Two Population Means: Matched Samples A Chicago-based firm has documents that must be A Chicago-based firm has documents that must be quickly distributed to district offices throughout the U.S. The firm must decide between two delivery services, UPX (United Parcel Express) and INTEX (International Express), to transport its documents.

37. 37 Slide © 2009 Thomson South-Western. All Rights Reserved n Example: Express Deliveries Inferences About the Difference Between Two Population Means: Matched Samples In testing the delivery times of the two services, the In testing the delivery times of the two services, the firm sent two reports to a random sample of its district offices with one report carried by UPX and the other report carried by INTEX. Do the data on the next slide indicate a difference in mean delivery times for the two services? Use a.05 level of significance.

38. 38 Slide © 2009 Thomson South-Western. All Rights Reserved 32 30 19 16 15 18 14 10 7 16 25 24 15 15 13 15 15 8 9 11 UPXINTEXDifference District Office Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver Delivery Time (Hours) 7 6 4 1 2 3 2 -2 5 Inferences About the Difference Between Two Population Means: Matched Samples

39. 39 Slide © 2009 Thomson South-Western. All Rights Reserved H 0 :  d = 0  H a :  d  Let  d = the mean of the difference values for the two delivery services for the population two delivery services for the population of district offices of district offices 1. Develop the hypotheses. Inferences About the Difference Between Two Population Means: Matched Samples p –Value and Critical Value Approaches p –Value and Critical Value Approaches

40. 40 Slide © 2009 Thomson South-Western. All Rights Reserved 2. Specify the level of significance.  =.05 Inferences About the Difference Between Two Population Means: Matched Samples p –Value and Critical Value Approaches p –Value and Critical Value Approaches 3. Compute the value of the test statistic.

41. 41 Slide © 2009 Thomson South-Western. All Rights Reserved 5. Determine whether to reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? We are at least 95% confident that there is a difference in mean delivery times for the two services? 4. Compute the p –value. For t = 2.94 and df = 9, the p –value is between For t = 2.94 and df = 9, the p –value is between.02 and.01. (This is a two-tailed test, so we double the upper-tail areas of.01 and.005.) Because p –value <  =.05, we reject H 0. Inferences About the Difference Between Two Population Means: Matched Samples p –Value Approach p –Value Approach

42. 42 Slide © 2009 Thomson South-Western. All Rights Reserved 4. Determine the critical value and rejection rule. Inferences About the Difference Between Two Population Means: Matched Samples Critical Value Approach Critical Value Approach For  =.05 and df = 9, t.025 = 2.262. Reject H 0 if t > 2.262 5. Determine whether to reject H 0. Because t = 2.94 > 2.262, we reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services?

43. 43 Slide © 2009 Thomson South-Western. All Rights Reserved End of Chapter 10 Part A