1. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Statistics for Business and Economics 7 th Edition Chapter 10 Hypothesis Testing: Additional Topics Ch. 10-1

2. Chapter Goals After completing this chapter, you should be able to: Test hypotheses for the difference between two population means Two means, matched pairs Independent populations, population variances known Independent populations, population variances unknown but equal Complete a hypothesis test for the difference between two proportions (large samples) Use the chi-square distribution for tests of the variance of a normal distribution Use the F table to find critical F values Complete an F test for the equality of two variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-2

3. Two Sample Tests Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Two Sample Tests Population Means, Independent Samples Population Means, Dependent Samples Population Variances Group 1 vs. independent Group 2 Same group before vs. after treatment Variance 1 vs. Variance 2 Examples: Population Proportions Proportion 1 vs. Proportion 2 Ch. 10-3

4. Dependent Samples Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Assumptions: Both Populations Are Normally Distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Dependent Samples d i = x i - y i Ch. 10-4 10.1

5. Test Statistic: Dependent Samples The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: where Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall D 0 = hypothesized mean difference s d = sample standard dev. of differences n = the sample size (number of pairs) Dependent Samples Ch. 10-5

6. Decision Rules: Matched Pairs Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Lower-tail test: H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 0 Matched or Paired Samples  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t n-1,  Reject H 0 if t > t n-1,  Reject H 0 if t < -t n-1,    or t > t n-1,  Where has n - 1 d.f. Ch. 10-6

7. 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: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Matched Pairs 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 Ch. 10-7

8. Has the training made a difference in the number of complaints (at the  = 0.05 level)? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall - 4.2d = H 0 : μ x – μ y = 0 H 1 : μ x – μ y  0 Test Statistic: Critical Value = ± 2.776 d.f. = n − 1 = 4 Reject  /2 - 2.776 2.776 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. Matched Pairs: Solution Reject  /2 - 1.66  =.05 Ch. 10-8

9. Difference Between Two Means Different populations Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population Normally distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μ x – μ y Ch. 10-9 10.2

10. Difference Between Two Means Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples Test statistic is a z value Test statistic is a a value from the Student’s t distribution σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal (continued) Ch. 10-10

11. σ x 2 and σ y 2 Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples Assumptions:  Samples are randomly and independently drawn  both population distributions are normal  Population variances are known * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown Ch. 10-11

12. σ x 2 and σ y 2 Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples …and the random variable has a standard normal distribution When σ x 2 and σ y 2 are known and both populations are normal, the variance of X – Y is (continued) * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown Ch. 10-12

13. Test Statistic, σ x 2 and σ y 2 Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown The test statistic for μ x – μ y is: Ch. 10-13

14. Hypothesis Tests for Two Population Means Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Lower-tail test: H 0 : μ x  μ y H 1 : μ x < μ y i.e., H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x ≤ μ y H 1 : μ x > μ y i.e., H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x = μ y H 1 : μ x ≠ μ y i.e., H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 0 Two Population Means, Independent Samples Ch. 10-14

15. Decision Rules Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Two Population Means, Independent Samples, Variances Known Lower-tail test: H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 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 Ch. 10-15

16. σ x 2 and σ y 2 Unknown, Assumed Equal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed  Population variances are unknown but assumed equal * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Ch. 10-16

17. σ x 2 and σ y 2 Unknown, Assumed Equal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples (continued)  The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ  use a t value with (n x + n y – 2) degrees of freedom * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Ch. 10-17

18. Test Statistic, σ x 2 and σ y 2 Unknown, Equal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Where t has (n 1 + n 2 – 2) d.f., and The test statistic for μ x – μ y is: Ch. 10-18

19. Decision Rules Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Lower-tail test: H 0 : μ x – μ y  0 H 1 : μ x – μ y < 0 Upper-tail test: H 0 : μ x – μ y ≤ 0 H 1 : μ x – μ y > 0 Two-tail test: H 0 : μ x – μ y = 0 H 1 : μ x – μ y ≠ 0  /2  -t  -t  /2 tt t  /2 Reject H 0 if t < -t (n1+n2 – 2),  Reject H 0 if t > t (n1+n2 – 2),  Reject H 0 if t < -t (n1+n2 – 2),   or t > t (n1+n2 – 2),  Two Population Means, Independent Samples, Variances Unknown Ch. 10-19

20. 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Assuming both populations are approximately normal with equal variances, is there a difference in average yield (  = 0.05)? Ch. 10-20

21. Calculating the Test Statistic Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The test statistic is: Ch. 10-21

22. Solution 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: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence of a difference in means. t 0 2.0154-2.0154.025 Reject H 0.025 2.040 Ch. 10-22

23. σ x 2 and σ y 2 Unknown, Assumed Unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples Assumptions:  Samples are randomly and independently drawn  Populations are normally distributed  Population variances are unknown and assumed unequal * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Ch. 10-23

24. σ x 2 and σ y 2 Unknown, Assumed Unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population means, independent samples (continued) Forming interval estimates:  The population variances are assumed unequal, so a pooled variance is not appropriate  use a t value with degrees of freedom, where σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 assumed unequal Ch. 10-24

25. Test Statistic, σ x 2 and σ y 2 Unknown, Unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Where t has degrees of freedom: The test statistic for μ x – μ y is: Ch. 10-25

26. Two Population Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Goal: Test hypotheses for the difference between two population proportions, P x – P y Population proportions Assumptions: Both sample sizes are large, nP(1 – P) > 5 Ch. 10-26 10.3

27. Two Population Proportions The random variable is approximately normally distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population proportions (continued) Ch. 10-27

28. Test Statistic for Two Population Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population proportions The test statistic for H 0 : P x – P y = 0 is a z value: Where Ch. 10-28

29. Decision Rules: Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Population proportions Lower-tail test: H 0 : P x – P y  0 H 1 : P x – P y < 0 Upper-tail test: H 0 : P x – P y ≤ 0 H 1 : P x – P y > 0 Two-tail test: H 0 : P x – P y = 0 H 1 : P x – P y ≠ 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  Ch. 10-29

30. 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-30

31. The hypothesis test is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall H 0 : P M – P W = 0 (the two proportions are equal) H 1 : P M – P W ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: = 36/72 =.50 Women: = 31/50 =.62  The estimate for the common overall proportion is: Example: Two Population Proportions (continued) Ch. 10-31

32. Example: Two Population Proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The test statistic for P M – P W = 0 is: (continued).025 -1.961.96.025 -1.31 Decision: Do not reject H 0 Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes. Reject H 0 Critical Values = ±1.96 For  =.05 Ch. 10-32

33. Hypothesis Tests for Two Variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Tests for Two Population Variances F test statistic H 0 : σ x 2 = σ y 2 H 1 : σ x 2 ≠ σ y 2 Two-tail test Lower-tail test Upper-tail test H 0 : σ x 2  σ y 2 H 1 : σ x 2 < σ y 2 H 0 : σ x 2 ≤ σ y 2 H 1 : σ x 2 > σ y 2  Goal: Test hypotheses about two population variances The two populations are assumed to be independent and normally distributed Ch. 10-33 10.4

34. Hypothesis Tests for Two Variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Tests for Two Population Variances F test statistic The random variable Has an F distribution with (n x – 1) numerator degrees of freedom and (n y – 1) denominator degrees of freedom Denote an F value with 1 numerator and 2 denominator degrees of freedom by (continued) Ch. 10-34

35. Test Statistic Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Tests for Two Population Variances F test statistic The critical value for a hypothesis test about two population variances is where F has (n x – 1) numerator degrees of freedom and (n y – 1) denominator degrees of freedom Ch. 10-35

36. Decision Rules: Two Variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall rejection region for a two- tail test is: F 0  Reject H 0 Do not reject H 0 F0  /2 Reject H 0 Do not reject H 0 H 0 : σ x 2 = σ y 2 H 1 : σ x 2 ≠ σ y 2 H 0 : σ x 2 ≤ σ y 2 H 1 : σ x 2 > σ y 2 Use s x 2 to denote the larger variance. where s x 2 is the larger of the two sample variances Ch. 10-36

37. Example: F Test 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  = 0.10 level? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-37

38. F Test: Example Solution Form the hypothesis test: H 0 : σ x 2 = σ y 2 ( there is no difference between variances) H 1 : σ x 2 ≠ σ y 2 ( there is a difference between variances) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Degrees of Freedom: Numerator (NYSE has the larger standard deviation): n x – 1 = 21 – 1 = 20 d.f. Denominator: n y – 1 = 25 – 1 = 24 d.f. Find the F critical values for  =.10/2: Ch. 10-38

39. F Test: Example Solution The test statistic is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall  /2 =.05 Reject H 0 Do not reject H 0 H 0 : σ x 2 = σ y 2 H 1 : σ x 2 ≠ σ y 2 F = 1.256 is not in the rejection region, so we do not reject H 0 (continued) Conclusion: There is not sufficient evidence of a difference in variances at  =.10 F Ch. 10-39

40. Two-Sample Tests in EXCEL 2007 For paired samples (t test): Data | data analysis… | t-test: paired two sample for means For independent samples: Independent sample Z test with variances known: Data | data analysis | z-test: two sample for means For variances… F test for two variances: Data | data analysis | F-test: two sample for variances Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-40

41. Chapter Summary Compared two dependent samples (paired samples) Performed paired sample t test for the mean difference Compared two independent samples Performed z test for the differences in two means Performed pooled variance t test for the differences in two means Compared two population proportions Performed z-test for two population proportions Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 10-41

42. Chapter Summary Performed F tests for the difference between two population variances Used the F table to find F critical values Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall (continued) Ch. 10-42