1. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-1 6 th & 7 th Lesson Hypothesis Testing for Two Population Parameters

2. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-2 Hypothesis Tests for the Difference Between Two Means Testing Hypotheses about μ 1 – μ 2 Use the same situations discussed already: Standard deviations known or unknown Sample sizes  30 or not  30

3. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-3 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

4. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-4 Hypothesis tests for μ 1 – μ 2 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 Use a z test statistic Use s to estimate unknown σ, approximate with a z test statistic Use s to estimate unknown σ, use a t test statistic

5. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-5 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 The test statistic for μ 1 – μ 2 is: σ 1 and σ 2 known *

6. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-6 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, large samples The test statistic for μ 1 – μ 2 is: *

7. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-7 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, small samples Where t  /2 or t  has (n 1 + n 2 – 2) d.f., and *Assuming equal variances The test statistic for μ 1 – μ 2 is: *

8. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-8 Population means, independent samples σ 1 and σ 2 known σ 1 and σ 2 unknown, n 1 and n 2  30 σ 1 and σ 2 unknown, n 1 or n 2 < 30 σ 1 and σ 2 unknown, small samples Where t  /2 or t  has d.f., *Assuming different variances The test statistic for μ 1 – μ 2 is: *

9. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-9 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

10. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-10 Pooled s p t Test: Example 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)?

11. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-11 Calculating the Test Statistic The test statistic is:

12. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-12 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 of a difference in means. t 0 2.0154-2.0154.025 Reject H 0.025 2.040

13. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-13 The test statistic for d is Paired samples Where t  /2 has n - 1 d.f. and s d is: n is the number of pairs in the paired sample Hypothesis Testing for Paired Samples

14. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-14 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)

15. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-15 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

16. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-16  Has the training made a difference in the number of complaints (at the 0.01 level)? - 4.2d = H 0 : μ d = 0 H A :  μ d  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. Paired Samples: Solution Reject  /2 - 1.66  =.01

17. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-17 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : p 1  p 2 H A : p 1 < p 2 i.e., H 0 : p 1 – p 2  0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 ≤ p 2 H A : p 1 > p 2 i.e., H 0 : p 1 – p 2 ≤ 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 = p 2 H A : p 1 ≠ p 2 i.e., H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 ≠ 0

18. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-18 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 p 1 = p 2 and pool the two p estimates

19. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-19 Two Population Proportions Population proportions The test statistic for p 1 – p 2 is: (continued)

20. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-20 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : p 1 – p 2  0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 – p 2 ≤ 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 – p 2 = 0 H A : p 1 – p 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 

21. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-21 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

22. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-22 The hypothesis test is: H 0 : p 1 – p 2 = 0 (the two proportions are equal) H A : p 1 – p 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)

23. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 9-23 The test statistic for p 1 – p 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 proportions who will vote yes between men and women. Reject H 0 Critical Values = ±1.96 For  =.05