1. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-1 Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 10 Two-Sample Tests

2. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-2 Learning Objectives In this chapter, you learn how to use hypothesis testing for comparing the difference between:  The means of two independent populations with equal variances  The means of two independent populations with unequal variances  The means of two related populations  The proportions of two independent populations  The variances of two independent populations

3. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-3 Two-Sample Tests Overview Two Sample Tests Independent Population Means Means, Related Populations Independent Population Variances Group 1 vs. Group 2 Same group before vs. after treatment Variance 1 vs. Variance 2 Examples Independent Population Proportions Proportion 1vs. Proportion 2

4. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-4 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown  Different data sources  Independent: Sample selected from one population has no effect on the sample selected from the other population  Use pooled variance t test, or separate-variance t test

5. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-5 The Process There are two tests to determine if the means of two populations are different. One test assumes the variances of the populations are equal the other is needed if the variances are not equal. Rather than make assumptions about the variances, I prefer to perform an F test for the equality of population variances and use the results to determine which model should be used. Therefore the F test comes first.

6. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-6 Testing Population Variances  Purpose: To determine if two independent populations have the same variability. H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 H 0 : σ 1 2  σ 2 2 H 1 : σ 1 2 < σ 2 2 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 Two-tail test Lower-tail testUpper-tail test

7. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-7 Testing Population Variances The F test statistic is: = Variance of Sample 1 n 1 - 1 = numerator degrees of freedom n 2 - 1 = denominator degrees of freedom = Variance of Sample 2

8. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-8 F Test: An Example You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data : NYSE NASDAQ Number 2125 Mean3.272.53 Std dev1.301.16 Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.05 level?

9. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-9 F Test: Example Solution  The test statistic is: H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2  The p-value is not < alpha, so we do not reject H 0 (continued)  Conclusion: There is not sufficient evidence of a difference in the variances of dividend yield for the two exchanges  p-value=.588846  =.05

10. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-10 F test in PhStat  PHStat | Two-Sample Tests | F Test for Differences in Two Variances …

11. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-11 Hypothesis Tests for Two Population Means Lower tail test: H 0 : μ 1  μ 2 H 1 : μ 1 < μ 2 i.e., H 0 : μ 1 – μ 2  0 H 1 : μ 1 – μ 2 < 0 Upper tail test: H 0 : μ 1 ≤ μ 2 H 1 : μ 1 > μ 2 i.e., H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Two-tailed test: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 i.e., H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0 Two Population Means, Independent Samples

12. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-12 Two-Sample Tests for Means Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed  or large sample sizes are used

13. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-13 Difference Between Two Means Population means, independent samples σ 1 and σ 2 unknown but equal σ 1 and σ 2 unknown but unequal  Use s to estimate unknown σ  Use pooled variance t test where  1 =  2 or  individual variance t test where  1  2 (proven by F test)

14. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-14 Population means, independent samples σ 1 and σ 2 Unknown but Equal If an F test does not prove the variances to be unequal, we can assume that they are equal and pool them. The pooled standard deviation is (continued) σ 1 and σ 2 unknown but equal

15. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-15 Population means, independent samples σ 1 and σ 2 Unknown but Equal Where t has (n 1 + n 2 – 2) d.f., and The test statistic for μ 1 = μ 2 is: σ 1 and σ 2 unknown but equal (continued)

16. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-16 Pooled Variance t Test: Example You are 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 Is there a difference in the mean dividend yield between the NYSE & NASDAQ (  = 0.05)?

17. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-17 Solution H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2  = 0.05 df = 21 + 25 - 2 = 44 Pooled Test Statistic: p-value: Decision: Conclusion: Reject H 0 at  = 0.05 There is sufficient evidence of a difference in mean dividend yield for the two exchanges. t 0.025 Reject H 0.025.047407 F = 1.256 p =.5888 Do Not Reject σ 1 2 = σ 2 2

18. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-18 Pooled Variance t Test in PhStat and Excel  If the raw data is available  Tools | Data Analysis | t-Test: Two Sample Assuming Equal Variances  If only summary statistics are available  PHStat | Two-Sample Tests | t Test for Differences in Two Means …

19. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-19 Independent Populations Unequal Variance  If you cannot assume population variances are equal, the pooled-variance t test is inappropriate  Instead, use a separate-variance t test, which includes the two separate sample variances in the computation of the test statistic  The computations are not in the Text but are as follows:

20. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-20 The Unequal-Variance t Test Population means, independent samples σ 1 and σ 2 unknown and unequal

21. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-21 Unequal Variance t Test: Example You are 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 2.30 1.16 Is there a difference in the mean dividend yield between the NYSE & NASDAQ (  = 0.05)?

22. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-22 Unequal Variance Solution H 0 :  1 =  2 H 1 :  1   2  = 0.05 df = 92 Test Statistics: Decision: Conclusion: Fail to Reject at  = 0.05 There is not sufficient evidence of a difference in mean dividend yields for the two exchanges t 0.025 Reject H 0 0.025.1901 p-value =.1901 t = 1.34 F = 3.9313 p =.001794 Reject H O :

23. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-23 Unequal Variance t Test in PhStat and Excel  If the raw data is available  Tools | Data Analysis | t-Test: Two Sample Assuming Unequal Variances  If only summary statistics are available  Use the Excel spreadsheet downloaded from the Homework web page

24. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-24 Confidence Intervals Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown The confidence interval for μ 1 – μ 2 is: Where

25. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-25 Two-Sample Tests Related Populations D = X 1 - X 2 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 have large samples

26. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-26 If σ D is unknown, we can estimate the unknown population standard deviation with a sample standard deviation. The test statistic for D is now a t statistic, with n-1 d.f.: Paired samples Mean Difference, σ D Unknown (continued)

27. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-27  Assume you send your salespeople to a “customer service” training workshop to reduce complaints. 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

28. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-28  Has the training made a difference in the number of complaints (at the 0.05 level)? - 4.2D = H 0 : μ D = 0 H 1 :  μ D  0 Test Statistic: d.f. = n - 1 = 4 Reject  /2 Decision: Do not reject H 0 (p-value is not <.05) Conclusion: There is not sufficient evidence of a change in the number of complaints. Paired Samples: Solution Reject  /2 -1.66  =.01 p-value=.1730

29. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-29 Related Sample t Test in PhStat and Excel  If the raw data is available  Tools | Data Analysis | t-Test: Paired Two Sample for Means  If only summary statistics are available  No test available

30. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-30 Two Population Proportions Goal: Test a hypothesis or form a confidence interval for two independent population proportions, π 1 = π 2 Assumptions: n 1 π 1  5, n 1 (1-π 1 )  5 n 2 π 2  5, n 2 (1-π 2 )  5

31. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-31 Two Population Proportions Hypothesis for Population Proportions Lower-tail test: H 0 : π 1  π 2 H 1 : π 1 < π 2 i.e., H 0 : π 1 – π 2  0 H 1 : π 1 – π 2 < 0 Upper-tail test: H 0 : π 1 ≤ π 2 H 1 : π 1 > π 2 i.e., H 0 : π 1 – π 2 ≤ 0 H 1 : π 1 – π 2 > 0 Two-tail test: H 0 : π 1 = π 2 H 1 : π 1 ≠ π 2 i.e., H 0 : π 1 – π 2 = 0 H 1 : π 1 – π 2 ≠ 0

32. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-32 Two Population Proportions Since you begin by assuming the null hypothesis is true, you assume π 1 = π 2 and pool the two sample (p) estimates. The pooled estimate for the overall proportion is: where X 1 and X 2 are the number of successes in samples 1 and 2

33. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-33 Two Population Proportions The test statistic for π 1 =  2 is a Z statistic: where

34. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-34 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 for a significant difference at the.05 level of significance

35. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-35 Check the Assumptions The sample proportions are:  Men: p s1 = 36/72 =.50  Women: p s2 = 31/50 =.62 Assumptions Men np=72(.5)=36 n(1-p)=72(.5)=36 Women np=50(.62)=31 n(1-p)50(.38)=19 All values > 5

36. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-36 Two Independent Population Proportions: Example  H 0 : π 1 = π 2 (the two proportions are equal)  H 1 : π 1  π 2  The sample proportions are:  Men: p 1 = 36/72 =.50  Women: p 2 = 31/50 =.62  The pooled estimate for the overall proportion is:

37. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-37 Two Independent Population Proportions: Example The test statistic for π 1 = π 2 is: P-value for -1.31 is.1902 At  =.05.025 -1.961.96.025 -1.31 Decision: Do not reject H 0 There is not sufficient evidence of a difference in the proportions of men and women who will vote yes on Proposition A. Reject H 0

38. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-38 Confidence Interval - Two Independent Population Proportions The confidence interval for π 1 – π 2 is:

39. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-39 Two Sample Tests in PHStat

40. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-40 Sample PHStat Output Input Output

41. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-41 Sample PHStat Output Input Output (continued)

42. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-42 Chapter Summary In this chapter, we have  Compared two independent samples  Performed pooled variance t test for the differences in two means  Examined unequal variance t test for the differences in two means  Formed confidence intervals for the differences between two means  Compared two related samples (paired samples)  Performed paired sample t test for the mean difference  Formed confidence intervals for the paired difference

43. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-43 Chapter Summary  Compared two population proportions  Performed Z-test for equality of two population proportions  Formed confidence intervals for the difference between two population proportions  Performed F tests for the difference between two population variances In this chapter, we have