1. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-1 Business Statistics, 4e by Ken Black Chapter 10 Statistical Inferences about Two Populations

2. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-2 Learning Objectives Test hypotheses and construct confidence intervals about the difference in two population means using the Z statistic. Test hypotheses and construct confidence intervals about the difference in two population means using the t statistic.

3. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-3 Learning Objectives Test hypotheses and construct confidence intervals about the difference in two related populations. Test hypotheses and construct confidence intervals about the differences in two population proportions. Test hypotheses and construct confidence intervals about two population variances.

4. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-4 Sampling Distribution of the Difference Between Two Sample Means Population 1 Population 2

5. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-5 Sampling Distribution of the Difference between Two Sample Means

6. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-6 Z Formula for the Difference in Two Sample Means When  1 2 and  2 2 are known and Independent Samples

7. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-7 Hypothesis Testing for Differences Between Means: The Wage Example (part 1) Rejection Region Non Rejection Region Critical Values Rejection Region H H o a : : 12 12 0 0    

8. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-8 Hypothesis Testing for Differences Between Means: The Wage Example (part 2) Rejection Region Non Rejection Region Critical Values Rejection Region 0

9. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-9 Hypothesis Testing for Differences Between Means: The Wage Example (part 3) Advertising Managers 74.25657.79171.115 96.23465.14567.574 89.80796.76759.621 93.26177.24262.483 103.03067.05669.319 74.19564.27635.394 75.93274.19486.741 80.74265.36057.351 39.67273.904 45.65254.270 93.08359.045 63.38468.508 Auditing Managers 69.96277.13643.649 55.05266.03563.369 57.82854.33559.676 63.36242.49454.449 37.19483.84946.394 99.19867.16071.804 61.25437.38672.401 73.06559.50556.470 48.03672.79067.814 60.05371.35171.492 66.35958.653 61.26163.508

10. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-10 Hypothesis Testing for Differences between Means: The Wage Example (part 4) Rejection Region Non Rejection Region Critical Values Rejection Region 0

11. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-11 Difference Between Means: Using Excel z-Test: Two Sample for Means Adv MgrAuditing Mgr Mean70.700162.187 Known Variance264.164166.411 Observations3234 Hypothesized Mean Difference0 z2.35 P(Z<=z) one-tail0.0094 z Critical one-tail1.64 P(Z<=z) two-tail0.0189 z Critical two-tail1.960

12. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-12 Demonstration Problem 10.1 (part 1) Non Rejection Region Critical Value Rejection Region 0

13. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-13 Demonstration Problem 10.1 (part 2) Non Rejection Region Critical Value Rejection Region 0

14. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-14 Confidence Interval to Estimate  1 -  2 When  1,  2 are known

15. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-15 Demonstration Problem 10.2

16. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-16 The t Test for Differences in Population Means Each of the two populations is normally distributed. The two samples are independent. The values of the population variances are unknown. The variances of the two populations are equal.  1 2 =  2 2

17. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-17 t Formula to Test the Difference in Means Assuming  1 2 =  2 2

18. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-18 Hernandez Manufacturing Company (part 1) Rejection Region Non Rejection Region Critical Values Rejection Region 0

19. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-19 Hernandez Manufacturing Company (part 2) Training Method A 56 5145 47 5243 42 5352 5042 48 4744 Training Method B 59 52 53 54 57 56 55 64 53 65 53 57

20. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-20 Hernandez Manufacturing Company (part 3)

21. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-21 MINITAB Output for Hernandez New-Employee Training Problem Twosample T for method A vs method B NMeanStDevSE Mean method A1547.734.421.1 method B1256.604.271.2 95% C.I. for mu method A - mu method B: (-12.2, -5.3) T-Test mu method A = mu method B (vs not =): T = -5.20 P=0.0000 DF = 25 Both use Pooled StDev = 4.35

22. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-22 EXCEL Output for Hernandez New-Employee Training Problem t-Test: Two-Sample Assuming Equal Variances Variable 1Variable 2 Mean 4 7.7356.5 Variance 19.495 18.27 Observations1512 Pooled Variance 18.957 Hypothesized Mean Difference0 df25 t Stat - 5.20 P(T<=t) one-tail1.12E-05 t Critical one-tail1.71 P(T<=t) two-tail2.23E-05 t Critical two-tail 2.06

23. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-23 Confidence Interval to Estimate  1 -  2 when  1 2 and  2 2 are unknown and  1 2 =  2 2

24. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-24 Dependent Samples Before and after measurements on the same individual Studies of twins Studies of spouses Individual 12345671234567 Before 32 11 21 17 30 38 14 After 39 15 35 13 41 39 22

25. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-25 Formulas for Dependent Samples

26. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-26 P/E Ratios for Nine Randomly Selected Companies Company2001 P/E Ratio2002 P/E Ratio 18.912.7 238.145.4 343.010.0 434.027.2 534.522.8 615.224.1 720.332.3 819.940.1 961.9106.5

27. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-27 Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies Rejection Region Non Rejection Region Critical Value 0 Rejection Region

28. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-28 Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies Company 2001 P/E Ratio 2002 P/E Ratiod 18.912.7-3.8 238.145.4-7.3 343.010.033.0 434.027.26.8 534.522.811.7 615.224.1-8.9 720.332.3-12.0 819.940.1-20.2 961.9106.5-44.6

29. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-29 Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies

30. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-30 Hypothesis Testing with Dependent Samples: P/E Ratios for Nine Companies t-Test: Paired Two Sample for Means 2001 P/E Ratio 2002 P/E Ratio Mean30.6435.68 Variance268.1837.5 Observations99 Pearson Correlation0.674 Hypothesized Mean Difference0 df8 t Stat-0.7 P(T<=t) one-tail0.252 t Critical one-tail1.86 P(T<=t) two-tail0.504 t Critical two-tail2.306

31. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-31 Hypothesis Testing with Dependent Samples: Demonstration Problem 10.5 Individual 12345671234567 Before 32 11 21 17 30 38 14 After 39 15 35 13 41 39 22 d -7 -4 -14 4 -11 -8

32. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-32 Hypothesis Testing with Dependent Samples: Demonstration Problem 10.5 Rejection Region Non Rejection Region Critical Value 0

33. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-33 Hypothesis Testing with Dependent Samples: Demonstration Problem 10.5

34. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-34 Confidence Intervals for Mean Difference for Related Samples

35. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-35 Difference in Number of New-House Sales RealtorMay 2001May 2002d 1811-3 21930-11 356 4913-4 535-2 604-4 71315-2 81117-6 9912-3 10512-7 11862 1225-3 1311101 14 22-8 1578 161215-3 17612-6 1810 0

36. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-36 Confidence Interval for Mean Difference in Number of New-House Sales

37. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-37 Sampling Distribution of Differences in Sample Proportions

38. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-38 Z Formula for the Difference in Two Population Proportions

39. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-39 Z Formula to Test the Difference in Population Proportions

40. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-40 Testing the Difference in Population Proportions (Demonstration Problem 10.6) Rejection Region Non Rejection Region Critical Values Rejection Region 0

41. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-41 Testing the Difference in Population Proportions (Demonstration Problem 10.6)

42. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-42 Confidence Interval to Estimate p 1 - p 2

43. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-43 Example Problem: When do men shop for groceries?

44. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-44 F Test for Two Population Variances

45. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-45 F Distribution with 1 = 10 and 2 = 8

46. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-46 A Portion of the F Distribution Table for  = 0.025 Numerator Degrees of Freedom Denominator Degrees of Freedom 123456789 1647.79799.48864.15899.60921.83937.11948.20956.64963.28 238.5139.0039.1739.2539.3039.3339.3639.3739.39 317.4416.0415.4415.1014.8814.7314.6214.5414.47 412.2210.659.989.609.369.209.078.988.90 510.018.437.767.397.156.986.856.766.68 68.817.266.606.235.995.825.705.605.52 78.076.545.895.525.295.124.994.904.82 87.576.065.425.054.824.654.534.434.36 97.215.715.084.724.484.324.204.104.03 106.945.464.834.474.244.073.953.853.78 116.725.264.634.284.043.883.763.663.59 126.555.104.474.123.893.733.613.513.44

47. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-47 Sheet Metal Example: Hypothesis Test for Equality of Two Population Variances (Part 1)

48. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-48 Sheet metal Manufacturer (Part 2) Rejection Regions Critical Values Non Rejection Region

49. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 10-49 Sheet Metal Example (Part 3) Machine 1 22.3 21.822.2 21.8 21.921.6 22.3 22.4 21.622.5 Machine 2 22.0 22.1 21.8 21.9 22.2 22.0 21.7 21.9 22.0 22.1 21.9 22.1