Business Research Methods William G. Zikmund

1. Business Research MethodsWilliam G. Zikmund Chapter 22: Bivariate Analysis - Tests of Differences

2. Common Bivariate Tests Type of Measurement Differences between two independent groups Differences among three or more independent groups Interval and ratio Independent groups: t-test or Z-test One-way ANOVA

3. Common Bivariate Tests Type of Measurement Differences between two independent groups Differences among three or more independent groups Ordinal Mann-Whitney U-test Wilcoxon test Kruskal-Wallis test

4. Common Bivariate Tests Type of Measurement Differences between two independent groups Differences among three or more independent groups Nominal Z-test (two proportions) Chi-square test Chi-square test

5. Type of Measurement Differences between two independent groups Nominal Chi-square test

6. Differences Between Groups • Contingency Tables • Cross-Tabulation • Chi-Square allows testing for significant differences between groups • “Goodness of Fit”

7. Chi-Square Test x² = chi-square statistics Oi = observed frequency in the ith cell Ei = expected frequency on the ith cell

8. Ri = total observed frequency in the ith row Cj = total observed frequency in the jth column n = sample size Chi-Square Test

9. Degrees of Freedom (R-1)(C-1)=(2-1)(2-1)=1

10. Degrees of Freedom d.f.=(R-1)(C-1)

11. Awareness of Tire Manufacturer’s Brand Men Women Total Aware 50 10 60 Unaware 1525 40 65 35 100

12. Chi-Square Test: Differences Among Groups Example

14. X2=3.84 with 1 d.f.

15. Type of Measurement Differences between two independent groups Interval and ratio t-test or Z-test

16. Differences Between Groups when Comparing Means • Ratio scaled dependent variables • t-test • When groups are small • When population standard deviation is unknown • z-test • When groups are large

17. Null Hypothesis About Mean Differences Between Groups

18. t-Test for Difference of Means

19. X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means

20. t-Test for Difference of Means

21. t-Test for Difference of Means X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means.

22. Pooled Estimate of the Standard Error

23. Pooled Estimate of the Standard Error S12 = the variance of Group 1 S22 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2

24. Pooled Estimate of the Standard Errort-test for the Difference of Means S12 = the variance of Group 1 S22 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2

25. Degrees of Freedom • d.f. = n - k • where: • n = n1 + n2 • k = number of groups

26. t-Test for Difference of Means Example

28. Type of Measurement Differences between two independent groups Nominal Z-test (two proportions)

29. Comparing Two Groups when Comparing Proportions • Percentage Comparisons • Sample Proportion - P • Population Proportion -

30. Differences Between Two Groups when Comparing Proportions The hypothesis is: Ho: P1 = P2 may be restated as: Ho: P1 - P2 = 0

31. Z-Test for Differences of Proportions or

32. Z-Test for Differences of Proportions

33. Z-Test for Differences of Proportions p1= sample portion of successes in Group 1 p2= sample portion of successes in Group 2 (p1 - p1)= hypothesized population proportion 1 minus hypothesized population proportion 1 minus Sp1-p2= pooled estimate of the standard errors of difference of proportions

34. Z-Test for Differences of Proportions

35. Z-Test for Differences of Proportions p= pooled estimate of proportion of success in a sample of both groups p= (1- p) or a pooled estimate of proportion of failures in a sample of both groups n1= sample size for group 1 n2= sample size for group 2

36. Z-Test for Differences of Proportions

37. Z-Test for Differences of Proportions

38. A Z-Test for Differences of Proportions

39. Differences between three or more independent groups Type of Measurement Interval or ratio One-way ANOVA

40. Analysis of Variance Hypothesis when comparing three groups m1 = m2 = m3

41. Analysis of Variance F-Ratio

42. Analysis of Variance Sum of Squares

43. Analysis of Variance Sum of SquaresTotal

44. pi= individual scores, i.e., the ith observation or test unit in the jth group pi= grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) Analysis of Variance Sum of Squares

45. Analysis of Variance Sum of SquaresWithin

46. pi= individual scores, i.e., the ith observation or test unit in the jth group pi= grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) Analysis of Variance Sum of SquaresWithin

47. Analysis of Variance Sum of Squares Between

48. = individual scores, i.e., the ith observation or test unit in the jth group = grand mean nj= number of all observations or test units in a group Analysis of Variance Sum of squares Between

49. Analysis of Variance Mean Squares Between

50. Analysis of Variance Mean Square Within