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  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 Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown Goal: Test hypothesis or form a confidence interval for the difference between two population means, μ 1 – μ 2 The point estimate for the difference between sample means: X 1 – X 2

5. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-5 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 the difference between 2 sample means  Use Z test, pooled variance t test, or separate-variance t test

6. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-6 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown Use a Z test statistic Use S to estimate unknown σ, use a t test statistic

7. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-7 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown Assumptions:  Samples are randomly and independently drawn  population distributions are normal

8. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-8 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown When σ 1 and σ 2 are known and both populations are normal, the test statistic is a Z-value and the standard error of X 1 – X 2 is

9. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-9 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown The test statistic is:

10. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-10 Two-Sample Tests Independent Populations 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-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 Independent Populations, Comparing Means

11. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-11 Two-Sample Tests Independent Populations Two Independent Populations, Comparing Means Lower-tail test: H 0 : μ 1 – μ 2  0 H 1 : μ 1 – μ 2 < 0 Upper-tail test: H 0 : μ 1 – μ 2 ≤ 0 H 1 : μ 1 – μ 2 > 0 Two-tail test: H 0 : μ 1 – μ 2 = 0 H 1 : μ 1 – μ 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if Z < -Z a Reject H 0 if Z > Z a Reject H 0 if Z < -Z a/2 or Z > Z a/2

12. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-12 Two-Sample Tests 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  Population variances are unknown but assumed equal

13. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-13 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown Forming interval estimates:  The population variances 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

14. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-14 Two-Sample Tests Independent Populations Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown The pooled standard deviation is:

15. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-15 Two-Sample Tests Independent Populations Where t has (n 1 + n 2 – 2) d.f., and The test statistic is: Independent Population Means σ 1 and σ 2 known σ 1 and σ 2 unknown

16. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-16 Two-Sample Tests Independent Populations  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 Assuming both populations are approximately normal with equal variances, is there a difference in average yield (  = 0.05)?

17. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-17 Two-Sample Tests Independent Populations The test statistic is:

18. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-18 Two-Sample Tests Independent Populations  H 0 : μ 1 - μ 2 = 0 i.e. (μ 1 = μ 2 )  H 1 : 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: 2.040 t 0 2.0154-2.0154.025 Reject H 0.025 Decision: Reject H 0 at α = 0.05 2.040 Conclusion: There is evidence of a difference in the 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 complicated and are best performed using Excel

20. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-20 Two-Sample Tests 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:

21. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-21 Two-Sample Tests 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

22. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-22 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

23. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-23 Two-Sample Tests Related Populations The ith paired difference is D i, where D i = X 1i - X 2i The point estimate for the population mean paired difference is D : Suppose the population standard deviation of the difference scores, σ D, is known.

24. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-24 Two-Sample Tests Related Populations The test statistic for the mean difference is a Z value: Where μ D = hypothesized mean difference σ D = population standard deviation of differences n = the sample size (number of pairs)

25. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-25 Two-Sample Tests Related Populations If σ D is unknown, you can estimate the unknown population standard deviation with a sample standard deviation:

26. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-26 Two-Sample Tests Related Populations The test statistic for D is now a t statistic: Where t has n - 1 d.f. and S D is:

27. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-27 Two-Sample Tests Related Populations Lower-tail test: H 0 : μ D  0 H 1 : μ D < 0 Upper-tail test: H 0 : μ D ≤ 0 H 1 : μ D > 0 Two-tail test: H 0 : μ D = 0 H 1 : μ D ≠ 0  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t a Reject H 0 if t > t a Reject H 0 if t < -t a/2 or t > t a/2

28. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-28 Two-Sample Tests Related Populations Example Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data: SalespersonNumber of ComplaintsDifference, D i (2-1) Before (1)After (2) C.B.64-2 T.F.206-14 M.H.32 R.K.000 M.O40-4

29. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-29 Two-Sample Tests Related Populations Example SalespersonNumber of ComplaintsDifference, D i (2-1) Before (1)After (2) C.B.64-2 T.F.206-14 M.H.32 R.K.000 M.O40-4

30. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-30 Two-Sample Tests Related Populations Example Has the training made a difference in the number of complaints (at the α = 0.01 level)? H 0 : μ D = 0 H 1 : μ D  0 Critical Value = ± 4.604 d.f. = n - 1 = 4 Test Statistic:

31. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-31 Two-Sample Tests Related Populations Example Reject - 4.604 4.604 Reject  /2 - 1.66 Decision: Do not reject H 0 (t statistic is not in the reject region) Conclusion: There is no evidence of a significant change in the number of complaints  /2

32. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-32 Two-Sample Tests Related Populations  The confidence interval for μ D (σ known) is: Where n = the sample size (number of pairs in the paired sample)

33. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-33 Two-Sample Tests Related Populations  The confidence interval for μ D (σ unknown) is: where

34. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-34 Two Population Proportions Goal: Test a hypothesis or form a confidence interval for the difference between 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 The point estimate for the difference is p 1 - p 2

35. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-35 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

36. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-36 Two Population Proportions The test statistic for p 1 – p 2 is a Z statistic: where

37. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-37 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

38. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-38 Two Population Proportions Hypothesis for Population Proportions Lower-tail test: H 0 : π 1 – π 2  0 H 1 : π 1 – π 2 < 0 Upper-tail test: H 0 : π 1 – π 2 ≤ 0 H 1 : π 1 – π 2 > 0 Two-tail test: H 0 : π 1 – π 2 = 0 H 1 : 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. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-39 Two Independent Population Proportions: Example  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 of 72 men, 36 indicated they would vote Yes and, in a sample of 50 women, 31 indicated they would vote Yes  Test at the.05 level of significance

40. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-40 Two Independent Population Proportions: Example  H 0 : π 1 – π 2 = 0 (the two proportions are equal)  H 1 : 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:

41. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-41 Two Independent Population Proportions: Example The test statistic for π 1 – π 2 is: Critical Values = ±1.96 For  =.05.025 -1.961.96.025 -1.31 Decision: Do not reject H 0 Conclusion: There is no evidence of a significant difference in proportions who will vote yes between men and women. Reject H 0

42. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-42 Two Independent Population Proportions The confidence interval for π 1 – π 2 is:

43. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-43 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

44. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-44 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

45. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-45 Testing Population Variances  The F critical value is found from the F table  There are two appropriate degrees of freedom: numerator and denominator.  In the F table,  numerator degrees of freedom determine the column  denominator degrees of freedom determine the row

46. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-46 Testing Population Variances 0  FLFL Reject H 0 Do not reject H 0 H 0 : σ 1 2  σ 2 2 H 1 : σ 1 2 < σ 2 2 Reject H 0 if F < F L 0  FUFU Reject H 0 Do not reject H 0 H 0 : σ 1 2 ≤ σ 2 2 H 1 : σ 1 2 > σ 2 2 Reject H 0 if F > F U Lower-tail testUpper-tail test

47. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-47 Testing Population Variances rejection region for a two-tail test is: F 0  /2 Reject H 0 Do not reject H 0 FUFU H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 FLFL  /2 Two-tail test

48. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-48 Testing Population Variances To find the critical F values: 1. Find F U from the F table for n 1 – 1 numerator and n 2 – 1 denominator degrees of freedom. 2. Find F L using the formula: Where F U* is from the F table with n 2 – 1 numerator and n 1 – 1 denominator degrees of freedom (i.e., switch the d.f. from F U )

49. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-49 Testing Population Variances  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?

50. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-50 Testing Population Variances  Form the hypothesis test:  H 0 : σ 2 1 – σ 2 2 = 0 (there is no difference between variances)  H 1 : σ 2 1 – σ 2 2 ≠ 0 (there is a difference between variances)  Numerator:  n 1 – 1 = 21 – 1 = 20 d.f.  Denominator:  n 2 – 1 = 25 – 1 = 24 d.f. F U = F.025, 20, 24 = 2.33  Numerator:  n 2 – 1 = 25 – 1 = 24 d.f.  Denominator:  n 1 – 1 = 21 – 1 = 20 d.f. F L = 1/F.025, 24, 20 = 0.41 FU:FU:FL:FL:

51. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-51 Testing Population Variances  The test statistic is: 0  /2 =.025 F U =2.33 Reject H 0 Do not reject H 0 F L =0.41  /2 =.025 Reject H 0 F  F = 1.256 is not in the rejection region, so we do not reject H 0  Conclusion: There is insufficient evidence of a difference in variances at  =.05

52. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-52 Chapter Summary In this chapter, we have  Compared two independent samples  Performed Z test for the differences in two means  Performed pooled 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 Z and t tests for the mean difference  Formed confidence intervals for the paired difference  Performed separate-variance t test

53. Statistics for Managers Using Microsoft Excel, 5e © 2008 Prentice-Hall, Inc.Chap 10-53 Chapter Summary  Compared two population proportions  Formed confidence intervals for the difference between two population proportions  Performed Z-test for two population proportions  Performed F tests for the difference between two population variances  Used the F table to find F critical values In this chapter, we have