1. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-1 Business Statistics: A Decision-Making Approach 7 th Edition Chapter 10 Estimation and Hypothesis Testing for Two Population Parameters

2. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-2 Chapter Goals After completing this chapter, you should be able to: Test hypotheses or form interval estimates for two independent population means Standard deviations known Standard deviations unknown two means from paired samples the difference between two population proportions

3. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-3 Estimation for Two Populations Estimating two population values Population means, independent samples Paired samples Population proportions Group 1 vs. independent Group 2 Same group before vs. after treatment Proportion 1 vs. Proportion 2 Examples:

4. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-4 Difference Between Two Means Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal Goal: Form a confidence interval for the difference between two population means, μ 1 – μ 2 The point estimate for the difference is x 1 – x 2 *

5. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-5 Independent Samples Population means, independent samples σ 1 and σ 2 known Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population Use the difference between 2 sample means * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

6. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-6 Population means, independent samples σ 1 and σ 2 known Assumptions:  Samples are randomly and independently drawn  population distributions are normal or both sample sizes are  30  Population standard deviations are known * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

7. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-7 Population means, independent samples σ 1 and σ 2 known…and the standard error of x 1 – x 2 is When σ 1 and σ 2 are known and both populations are normal or both sample sizes are at least 30, the test statistic is a z value… (continued) σ 1 and σ 2 known * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

8. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-8 Population means, independent samples σ 1 and σ 2 known The confidence interval for μ 1 – μ 2 is: σ 1 and σ 2 known (continued) * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

9. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-9 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, large samples Assumptions:  Samples are randomly and independently drawn  Population standard deviations are unknown  The two standard deviations are equal * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

10. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-10 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, large samples (continued) * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal Forming interval estimates:  The population standard deviations are assumed equal, so use the two sample standard deviations and pool them to estimate σ  the test statistic is a t value with (n 1 + n 2 – 2) degrees of freedom

11. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-11 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, large samples (continued) * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal The pooled standard deviation is

12. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-12 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, large samples (continued) * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal The confidence interval for μ 1 – μ 2 is: Where t  /2 has (n 1 + n 2 – 2) d.f., and

13. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-13 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, small samples Assumptions:  populations are normally distributed  there is a reason to believe that the populations do not have equal variances  samples are independent * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

14. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-14 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, small samples Forming interval estimates:  The population variances are not assumed equal, so we do not pool them  the test statistic is a t value with degrees of freedom given by: (continued) * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

15. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-15 Population means, independent samples σ 1 and σ 2 known The confidence interval for μ 1 – μ 2 is: σ 1 and σ 2 unknown, small samples (continued) Where t  /2 has d.f. given by * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

16. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-16 Hypothesis Tests for the Difference Between Two Means Testing Hypotheses about μ 1 – μ 2 Use the same situations discussed already: Standard deviations known Standard deviations unknown Assumed equal Assumed not equal

17. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-17 Hypothesis Tests for Two Population Proportions Lower tail test: H 0 : μ 1  μ 2 H A : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2  0 H A : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 ≤ μ 2 H A : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 H A : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 = μ 2 H A : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 ≠ 0 Two Population Means, Independent Samples

18. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-18 Hypothesis tests for μ 1 – μ 2 Population means, independent samples σ 1 and σ 2 known Use a z test statistic Use s p to estimate unknown σ, use a t test statistic with n 1 + n 2 – 2 d.f. Use s 1 and s 2 to estimate unknown σ 1 and σ 2, use a t test statistic and calculate the required degrees of freedom σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

19. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-19 Population means, independent samples σ 1 and σ 2 known The test statistic for μ 1 – μ 2 is: σ 1 and σ 2 known * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal

20. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-20 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, large samples * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal Where t has (n 1 + n 2 – 2) d.f., and The test statistic for μ 1 – μ 2 is:

21. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-21 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, small samples The test statistic for μ 1 – μ 2 is: * σ 1 and σ 2 unknown but assumed equal σ 1 and σ 2 unknown, not assumed equal Where t has d.f. given by

22. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-22 Two Population Means, Independent Samples Lower tail test: H 0 : μ 1 – μ 2  0 H A : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 – μ 2 ≤ 0 H A : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 – μ 2 = 0 H A : μ 1 – μ 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if z < -z  Reject H 0 if z > z  Reject H 0 if z < -z  /2  or z > z  /2 Hypothesis tests for μ 1 – μ 2 Example: σ 1 and σ 2 known:

23. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-23 Pooled s p t Test Example σ1 and σ2 unknown, assumed equal You’re a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data: NYSE NASDAQ Number 21 25 Sample mean 3.27 2.53 Sample std dev 1.30 1.16 Assuming equal variances, is there a difference in average yield (  = 0.05)?

24. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-24 Calculating the Test Statistic The test statistic is: Where:

25. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-25 Solution H 0 : μ 1 - μ 2 = 0 i.e. (μ 1 = μ 2 ) H A : μ 1 - μ 2 ≠ 0 i.e. (μ 1 ≠ μ 2 )  = 0.05 df = 21 + 25 - 2 = 44 Critical Values: t = ± 2.0154 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence that the means are different. t 0 2.0154-2.0154.025 Reject H 0.025 2.040

26. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-26 Paired Samples Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Or, if Not Normal, use large samples Paired samples d = x 1 - x 2

27. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-27 Paired Differences The i th paired difference is d i, where Paired samples d i = x 1i - x 2i The point estimate for the population mean paired difference is d : The sample standard deviation is n is the number of pairs in the paired sample

28. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-28 Paired Differences The confidence interval for d is Paired samples Where t has n - 1 d.f. and s d is: (continued) n is the number of pairs in the paired sample

29. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-29 The test statistic for d is Paired samples Where t has n - 1 d.f. and s d is: n is the number of pairs in the paired sample Hypothesis Testing for Paired Samples

30. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-30 Lower tail test: H 0 : μ d  0 H A : μ d < 0 Upper tail test: H 0 : μ d ≤ 0 H A : μ d > 0 Two-tailed test: H 0 : μ d = 0 H A : μ d ≠ 0 Paired Samples Hypothesis Testing for Paired Samples  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t  Reject H 0 if t > t  Reject H 0 if t < -t   or t > t  Where t has n - 1 d.f. (continued)

31. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-31 Assume you send your salespeople to a “customer service” training workshop. Is the training effective? You collect the following data: Paired Samples Example Number of Complaints: (2) - (1) Salesperson Before (1) After (2) Difference, d i C.B. 6 4 - 2 T.F. 20 6 -14 M.H. 3 2 - 1 R.K. 0 0 0 M.O. 4 0 - 4 -21 d =  didi n = -4.2

32. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-32  Has the training made a difference in the number of complaints (at the 0.05 level)? - 4.2d = H 0 : μ d = 0 H A :  μ d  0 Test Statistic: Critical Value = ± 2.7765 d.f. = n - 1 = 4 Reject  /2 - 2.7765 2.7765 Decision: Do not reject H 0 (t stat is not in the reject region) Conclusion: There is not a significant change in the number of complaints. Paired Samples: Solution Reject  /2 - 1.66  =.05

33. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-33 Two Population Proportions Goal: Form a confidence interval for or test a hypothesis about the difference between two population proportions, π 1 – π 2 The point estimate for the difference is p 1 – p 2 Population proportions Assumptions: n 1 π 1  5, n 1 (1-π 1 )  5 n 2 π 2  5, n 2 (1-π 2 )  5

34. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-34 Confidence Interval for Two Population Proportions Population proportions The confidence interval for π 1 – π 2 is:

35. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-35 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : π 1  π 2 H A : π 1 < π 2 i.e., H 0 : π 1 – π 2  0 H A : π 1 – π 2 < 0 Upper tail test: H 0 : π 1 ≤ π 2 H A : π 1 > π 2 i.e., H 0 : π 1 – π 2 ≤ 0 H A : π 1 – π 2 > 0 Two-tailed test: H 0 : π 1 = π 2 H A : π 1 ≠ π 2 i.e., H 0 : π 1 – π 2 = 0 H A : π 1 – π 2 ≠ 0

36. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-36 Two Population Proportions Population proportions The pooled estimate for the overall proportion is: where x 1 and x 2 are the numbers from samples 1 and 2 with the characteristic of interest Since we begin by assuming the null hypothesis is true, we assume π 1 = π 2 and pool the two p estimates

37. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-37 Two Population Proportions Population proportions The test statistic for π 1 – π 2 is: (continued)

38. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-38 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : π 1 – π 2  0 H A : π 1 – π 2 < 0 Upper tail test: H 0 : π 1 – π 2 ≤ 0 H A : π 1 – π 2 > 0 Two-tailed test: H 0 : π 1 – π 2 = 0 H A : π 1 – π 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if z < -z  Reject H 0 if z > z  Reject H 0 if z < -z   or z > z 

39. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-39 Example: Two population Proportions Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A? In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes Test at the.05 level of significance

40. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-40 The hypothesis test is: H 0 : π 1 – π 2 = 0 (the two proportions are equal) H A : π 1 – π 2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p 1 = 36/72 =.50 Women: p 2 = 31/50 =.62  The pooled estimate for the overall proportion is: Example: Two population Proportions (continued)

41. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-41 The test statistic for π 1 – π 2 is: Example: Two population Proportions (continued).025 -1.961.96.025 -1.31 Decision: Do not reject H 0 Conclusion: There is not significant evidence of a difference in the proportion who will vote yes between men and women. Reject H 0 Critical Values = ±1.96 For  =.05

42. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-42 Two Sample Tests in EXCEL For independent samples: Independent sample Z test with variances known: Data | data analysis | z-Test: Two Sample for Means Independent sample Z test with variance unknown: Data | data analysis | t-Test: Two Sample Assuming Equal Variances Data | data analysis | t-Test: Two Sample Assuming Unequal Variances For paired samples (t test): Data | data analysis… | t-Test: Paired Two Sample for Means

43. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-43 Two Sample Tests in PHStat Options

44. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-44 Two Sample Tests in PHStat Input Output

45. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-45 Two Sample Tests in PHStat Input Output

46. Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc. Chap 10-46 Chapter Summary Compared two independent samples Formed confidence intervals for the differences between two means Performed z test for the differences in two means Performed t test for the differences in two means Compared two related samples (paired samples) Formed confidence intervals for the paired difference Performed paired sample t tests for the mean difference Compared two population proportions Formed confidence intervals for the difference between two population proportions Performed z test for two population proportions