Narrated+Chapter+Ten.pptx. for thee marketing

hypotheses introduction

1. Chapter 10 – Two -Sample Hypothesis Tests Comparing Two Independent Samples: z and t- tests for Differences Between Two Population Means. Comparing Two Related ( Dependent ) Samples: z and t- tests for the Population Mean Difference. F- test for Differences Between Two Population Variances. “Rule-of-2”. Comparing Two Population Proportions ( z -test only). 1 BUSN 5760 - Business Statistics

2. Z -test for the Difference Between Two Population Means ( μ 1 - μ 2 ) Assume n 1 & n 2 ≥ 30 2 BUSN 5760 - Business Statistics Note: If σ isn’t given, substitute s

3. Hypothesis Testing for Differences Between Population Means: Two Wage Examples ($000s) Advertising Managers 74.256 57.791 71.115 96.234 65.145 67.574 89.807 96.767 59.621 93.261 77.242 62.483 103.030 67.056 69.319 74.195 64.276 35.394 75.932 74.194 86.741 80.742 65.360 57.351 39.672 73.904 45.652 54.270 93.083 59.045 63.384 68.508 Auditing Managers 69.962 77.136 43.649 55.052 66.035 63.369 57.828 54.335 59.676 63.362 42.494 54.449 37.194 83.849 46.394 99.198 67.160 71.804 61.254 37.386 72.401 73.065 59.505 56.470 48.036 72.790 67.814 60.053 71.351 71.492 66.359 58.653 61.261 63.508 3 BUSN 5760 - Business Statistics

4.  =0.05, /2 = 0.025, z 0.025 = 1.96 4 BUSN 5760 - Business Statistics Question: Is there a significant population mean difference between the two sample wages at α = 0.05 ? Hypothesis Testing for Differences Between Population Means: Two Wage Examples ($000s)

5. Since the observed value of 2.35 > 1.96, reject the null hypothesis. That is, there is a statistically significant population mean difference between the average annual wage of advertising managers and the average annual wage of auditing manager. 5 BUSN 5760 - Business Statistics 2.35 Hypothesis Testing for Differences Between Population Means: Two Wage Examples ($000s)

6. z-Test: Two Sample for Means   Adv Mgr Auditing Mgr Mean 70.7001 62.187 Known Variance 264.164 166.411 Observations 32 34 Hypothesized Mean Difference z 2.35 P(Z<=z) one-tail 0.0094 z Critical one-tail 1.64 P(Z<=z) two-tail 0.0189 z Critical two-tail 1.960   Difference Between Population Means: Using Excel p -value approach: If p < α , reject the null hypothesis Since p < α , reject the null hypothesis 6 BUSN 5760 - Business Statistics

7. Difference Between Population Means: Using PHStat2 (Input) p -value approach: If p < α , reject the null hypothesis 7 BUSN 5760 - Business Statistics Always “0”

8. Difference Between Population Means: Using PHStat2 (output) p -value approach: If p < α , reject the null hypothesis 8 BUSN 5760 - Business Statistics

9. Pooled- Variance (equal) t- Test for the Difference in Two Population Means (  1 -  2 ) , Assuming σ 1 2 ≈ σ 2 2 . More later. df = n 1 + n 2 - 2 Note: Excel (only) requires raw data! 9 BUSN 5760 - Business Statistics

10. “Sprague” Manufacturing Company Training Method A 56 51 45 47 52 43 42 53 52 50 42 48 47 44 44 Training Method B 59 52 53 54 57 56 55 64 53 65 53 57 10 BUSN 5760 - Business Statistics Question: is there a significant population mean difference between the two training methods at an α = 0.05 ? Assume “ equal ” (pooled) variances. More later.

11. 11 BUSN 5760 - Business Statistics H : μ A = μ B ( no significant difference in the population means) H a : μ A ≠ μ B ( is a significant difference in the population means) “Sprague” Manufacturing Company

12. EXCEL Output for “Sprague” New-Employee Training Problem t-Test: Two-Sample Assuming Equal Variances Variable A Variable B Mean 4 7.73 56.5 Variance 19.495 18.27 Observations 15 12 Pooled Variance 18.957 Hypothesized Mean Difference df 25 t Stat - 5.20 P(T<=t) one-tail 1.12E-05 t Critical one-tail 1.71 P(T<=t) two-tail 2.23E-05 t Critical two-tail 2.06 12 BUSN 5760 - Business Statistics P < α , reject H

13. PHStat2 Solution for “Sprague” New-Employee Training Problem 13 BUSN 5760 - Business Statistics

14. Separate -Variance (unequal) t- Test for the Difference in Two Population Means (  1 -  2 ) , Assuming σ 1 2 ≠ σ 2 2 where Note: Excel requires raw data! 14 BUSN 5760 - Business Statistics

15. “Sprague” Manufacturing Company Training Method A 23 51 95 99 13 13 42 99 7 88 89 88 17 14 13 Training Method B 59 52 53 54 57 56 55 64 53 65 53 57 15 BUSN 5760 - Business Statistics Question: is there now a significant population mean difference between the two training methods at an α = 0.05 ? Do Not Assume “ equal ” variances. More later. Notice completely different data sets.

16. Separate -Variance t- Test for the Difference in Two Population Means (  1 -  2 ) , Assuming σ 1 2 ≠ σ 2 2 where Note: Excel requires raw data! 16 BUSN 5760 - Business Statistics ± t α /2 = |2.1448|

17. PHStat2 New Solution for “Sprague” New-Employee Training Problem 17 BUSN 5760 - Business Statistics P > α , accept H o

18. where 18 BUSN 5760 - Business Statistics Paired z - test for the Population Mean Differences ( Dependent Samples – assuming the differences are normal, n ≥ 30)

19. where df = n – 1 ( n is the number of pairs of data) 19 BUSN 5760 - Business Statistics Paired t - test for the Population Mean Differences ( Dependent Samples – assuming the differences are normal)

20. “Sprague” Manufacturing Company Before Training Method 56 51 45 47 52 43 42 53 52 50 42 48 47 44 44 20 BUSN 5760 - Business Statistics Question: is there a significant population mean difference between “before” versus “after” a training method at an α = 0.05 ? Assume differences are normal. Notice: Same participants. After Training Method 56 60 65 57 62 43 52 63 52 60 52 58 47 54 54 H : μ d = 0. No significant population mean difference. H 1 : μ d ≠ 0. Is a significant population mean difference.

21. “Sprague” Manufacturing Company 21 BUSN 5760 - Business Statistics Question: is there a significant population mean difference between “before” versus “after” a training program at an α = 0.05 ? Assume differences are normal. PHStat2 Input.

22. “Sprague” Manufacturing Company 22 BUSN 5760 - Business Statistics Question: is there a significant population mean difference between “before” versus “after” a training program at an α = 0.05 ? Assume differences are normal. PHStat2 Output. H : μ d = 0. No significant population mean difference. H 1 : μ d ≠ 0. Is a significant population mean difference.

23. But How do We Know if the Two Variances Are “ Equal ” (pooled)? “The rule of 2” If the larger variance divided by the smaller variance is no larger than 2, the variances are said to be “equal” Note: Not met to be a substitute for the F -test (see next slides), but simply just a good “rule-of- thumb”. Note: Assumption: Populations are normally distributed. 23 BUSN 5760 - Business Statistics

24. F -Test for Differences Between Two Population Variances Where n 1 = size of sample taken from population 1 n 2 = size of sample taken from population 2 n 1 – 1 = degree of freedom from sample 1 (the numerator ( largest variance) degree of freedom n 2 – 1 = degree of freedom from sample 2 (the denominator ( smallest variance) degree of freedom) s 1 2 = variance of sample 1 ( largest ) s 2 2 = variance of sample 2 ( smallest ) 24 BUSN 5760 - Business Statistics

25. F -Distribution BUSN 5760 - Business Statistics 25

26. F -Distribution BUSN 5760 - Business Statistics 26

27. F -Test for Differences Between Two Population Variances Example: Back to Slide #3 Where n 1 = size of sample taken from population 1 = 32 n 2 = size of sample taken from population 2 = 34 n 1 – 1 = degree of freedom from sample 1 (the numerator ( largest variance ) degree of freedom = 31 n 2 – 1 = degree of freedom from sample 2 (the denominator ( smallest variance ) degree of freedom) = 33 s 1 2 = variance of sample 1 ( largest ) = 264.164 s 2 2 = variance of sample 2 ( smallest ) = 166.411 27 BUSN 5760 - Business Statistics

28. F -Test for Differences Between Two Population Variances Example: Back to Slide #3 28 BUSN 5760 - Business Statistics H : σ 1 2 = σ 2 2 ( Population variances are not different) H 1 : σ 1 2 ≠ σ 2 2 ( Population variances are different) Test at α = 0.10 (0.05 in upper -tail) F crit = 1.82 (see Slide # 30) Accept H . 1.82 1.59

29. F -Test for Differences Between Two Population Variances Example: Back to Slide #3 29 BUSN 5760 - Business Statistics

30. F -Distribution BUSN 5760 - Business Statistics 30

31. Z -test for the Difference Between Two Population Proportions ( π 1 - π 2 ) – PHStat2 Only 31 BUSN 5760 - Business Statistics

32. “Sprague” Manufacturing Company Without Training 56 51 45 47 52 43 42 53 52 50 42 48 47 44 44 32 BUSN 5760 - Business Statistics Question: Assuming that ≥ 50% is “passing”, is there a significant population proportional difference between Without Training versus With Training at an α = 0.05 ? Z -test only. With Training 56 60 65 57 62 43 52 63 52 60 52 58 47 54 54 H o : π Without = π With . No significant difference in the population proportions passing. H 1 : π Without ≠ π With . Is a significant difference in the population proportions passing.

33. Z -test for the Difference Between Two Population Proportions ( π 1 - π 2 ) – PHStat2 Only. 33 BUSN 5760 - Business Statistics H : π Without = π With . No significant difference in the population proportions passing. H 1 : π Without ≠ π With . Is a significant difference in the population proportions passing. Z test > |1.96|, Reject H .

34. Z -test for the Difference Between Two Population Proportions ( π 1 - π 2 ) – PHStat2 Only. 34 BUSN 5760 - Business Statistics H : π Without = π With . No significant difference in the population proportions passing. H 1 : π Without ≠ π With . Is a significant difference in the population proportions passing.

35. End! BUSN 5760 - Business Statistics 35