1. 1/45 Chapter 11 Hypothesis Testing II EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008

2. 2/45 Two Sample Tests Population Means, Independent Samples Population Means, Matched Pairs 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 (Note similarities to Chapter 9)

3. 3/45 Matched Pairs 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 Matched Pairs d i = x i - y i

4. 4/45 The test statistic for the mean difference is a t value, with n – 1 degrees of freedom: Test Statistic: Matched Pairs Where D 0 = hypothesized mean difference s d = sample standard dev. of differences n = the sample size (number of pairs) Matched Pairs

5. 5/45 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 Paired Samples Decision Rules: Matched Pairs  /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.

6. 6/45 Assume you send your sales people to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data: 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

7. 7/45 Has the training made a difference in the number of complaints (at the  = 0.01 level)? - 4.2d = H 0 : μ x – μ y = 0 H 1 : μ x – μ y  0 Test Statistic: Critical Value = ± 4.604 d.f. = n - 1 = 4 Reject  /2 - 4.604 4.604 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  =0.01

8. 8/45 Difference Between Two Means Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μ x – μ y Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population

9. 9/45 Difference Between Two Means Population means, independent samples Test statistic is a z value Test statistic is 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)

10. 10/45 Population means, independent samples σ x 2 and σ y 2 Known 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

11. 11/45 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 σ x 2 and σ y 2 Known

12. 12/45 Population means, independent samples Test Statistic, σ x 2 and σ y 2 Known * 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:

13. 13/45 Hypothesis Tests for Two Population Means 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

14. 14/45 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 Decision Rules

15. 15/45 Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Equal 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

16. 16/45 Population means, independent samples (continued) Forming interval estimates:  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 σ x 2 and σ y 2 Unknown, Assumed Equal

17. 17/45 * Test Statistic, σ x 2 and σ y 2 Unknown, Equal 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:

18. 18/45 Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Unequal 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

19. 19/45 Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Unequal (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

20. 20/45 * Test Statistic, σ x 2 and σ y 2 Unknown, Unequal 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:

21. 21/45 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 Decision Rules  /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 t has n - 1 d.f. Two Population Means, Independent Samples, Variances Unknown

22. 22/45 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 Assuming both populations are approximately normal with equal variances, is there a difference in average yield (  = 0.05)?

23. 23/45 Calculating the Test Statistic The test statistic is:

24. 24/45 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: 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

25. 25/45 Two Population Proportions 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) > 9

26. 26/45 Two Population Proportions Population proportions (continued) The random variable is approximately normally distributed

27. 27/45 Test Statistic for Two Population Proportions Population proportions The test statistic for H 0 : P x – P y = 0 is a z value: Where

28. 28/45 Decision Rules: Proportions 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 

29. 29/45 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 0.05 level of significance

30. 30/45 The hypothesis test is: 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 = 0.50 Women: = 31/50 = 0.62  The estimate for the common overall proportion is: Example: Two Population Proportions (continued)

31. 31/45 The test statistic for P M – P W = 0 is: Example: Two Population Proportions (continued) 0.025 -1.961.96 0.025 -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  = 0.05

32. 32/45 Population Variance follows a chi-square distribution with (n – 1) degrees of freedom  Goal: Test hypotheses about the population variance, σ 2  If the population is normally distributed, Hypothesis Tests of one Population Variance

33. 33/45 Confidence Intervals for the Population Variance Population Variance The test statistic for hypothesis tests about one population variance is (continued)

34. 34/45 Decision Rules: Variance Population variance Lower-tail test: H 0 : σ 2  σ 0 2 H 1 : σ 2 < σ 0 2 Upper-tail test: H 0 : σ 2 ≤ σ 0 2 H 1 : σ 2 > σ 0 2 Two-tail test: H 0 : σ 2 = σ 0 2 H 1 : σ 2 ≠ σ 0 2  /2  Reject H 0 if or

35. 35/45 Hypothesis Tests for Two Variances 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

36. 36/45 Hypothesis Tests for Two Variances 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 (continued)

37. 37/45 Test Statistic 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

38. 38/45 Decision Rules: Two Variances 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

39. 39/45 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?

40. 40/45 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) 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  = 0.10/2

41. 41/45 The test statistic is:  /2 =0.05 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 Test: Example Solution 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  = 0.10 F

42. 42/45 Two-Sample Tests in EXCEL For paired samples (t test): Tools | data analysis… | t-test: paired two sample for means For independent samples: Independent sample Z test with variances known: Tools | data analysis | z-test: two sample for means For variances… F test for two variances: Tools | data analysis | F-test: two sample for variances

43. 43/45 Two-Sample Tests in PHStat

44. 44/45 Sample PHStat Output Input Output

45. 45/45 Sample PHStat Output Input Output (continued)