1. Cross Tabulation and Chi Square Test for Independence

2. Cross-tabulation Helps answer questions about whether two or more variables of interest are linked: –Is the type of mouthwash user (heavy or light) related to gender? –Is the preference for a certain flavor (cherry or lemon) related to the geographic region (north, south, east, west)? –Is income level associated with gender? Cross-tabulation determines association not causality.

3. The variable being studied is called the dependent variable or response variable. A variable that influences the dependent variable is called independent variable. Dependent and Independent Variables

4. Cross-tabulation Cross-tabulation of two or more variables is possible if the variables are discrete: –The frequency of one variable is subdivided by the other variable categories. Generally a cross-tabulation table has: –Row percentages –Column percentages –Total percentages Which one is better? DEPENDS on which variable is considered as independent.

5. Cross tabulation

6. A contingency table shows the conjoint distribution of two discrete variables This distribution represents the probability of observing a case in each cell –Probability is calculated as: Contingency Table Observed cases Total cases P=

7. Chi-square Test for Independence The Chi-square test for independence determines whether two variables are associated or not. H 0 : Two variables are independent H 1 : Two variables are not independent Chi-square test results are unstable if cell count is lower than 5

8. x² = chi-square statistics O i = observed frequency in the i th cell E i = expected frequency on the i th cell R i = total observed frequency in the i th row C j = total observed frequency in the j th column n = sample size E ij = estimated cell frequency Estimated cell Frequency Chi-Square statistic Chi-Square Test Degrees of Freedom d.f.=(R-1)(C-1)

9. Aware50/39 10/21 60 Unaware 15/21 25/14 40 65 35 100 MenWomenTotal Awareness of Tire Manufacturer’s Brand

10. Chi-Square Test: Differences Among Groups Example X 2 with 1 d.f. at.05 critical value = 3.84

11. Chi-square Test for Independence Under H 0, the joint distribution is approximately distributed by the Chi- square distribution (  2 ).  22 Reject H 0 Chi-square 3.84 22.16

12. 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

13. Null Hypothesis About Mean Differences Between Groups

14. t-Test for Difference of Means

15. X 1 = mean for Group 1 X 2 = mean for Group 2 S X 1 -X 2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means

17. X 1 = mean for Group 1 X 2 = mean for Group 2 S X 1 -X 2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means

18. Pooled Estimate of the Standard Error

19. S 1 2 = the variance of Group 1 S 2 2 = the variance of Group 2 n 1 = the sample size of Group 1 n 2 = the sample size of Group 2 Pooled Estimate of the Standard Error

20. Pooled Estimate of the Standard Error t-test for the Difference of Means S 1 2 = the variance of Group 1 S 2 2 = the variance of Group 2 n 1 = the sample size of Group 1 n 2 = the sample size of Group 2

21. Degrees of Freedom d.f. = n - k where: –n = n 1 + n 2 –k = number of groups

22. t-Test for Difference of Means Example

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

25. Differences Between Two Groups when Comparing Proportions The hypothesis is: H o :  1   may be restated as: H o :  1   

26. or Z-Test for Differences of Proportions

28. p 1 = sample portion of successes in Group 1 p 2 = sample portion of successes in Group 2  1  1 )  = hypothesized population proportion 1 minus hypothesized population proportion 1 minus S p1-p2 = pooled estimate of the standard errors of difference of proportions Z-Test for Differences of Proportions

30. 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 n  = sample size for group 1 n  = sample size for group 2 Z-Test for Differences of Proportions

33. A Z-Test for Differences of Proportions

34. Analysis of Variance Hypothesis when comparing three groups  1    

35. Analysis of Variance F-Ratio

36. Analysis of Variance Sum of Squares

37. Analysis of Variance Sum of SquaresTotal

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

39. Analysis of Variance Sum of SquaresWithin

40. p i = individual scores, i.e., the i th observation or test unit in the j th group p i = grand mean n = number of all observations or test units in a group c = number of j th groups (or columns)

41. Analysis of Variance Sum of Squares Between

42. Analysis of Variance Sum of squares Between = individual scores, i.e., the i th observation or test unit in the j th group = grand mean n j = number of all observations or test units in a group

43. Analysis of Variance Mean Squares Between

44. Analysis of Variance Mean Square Within

45. Analysis of Variance F-Ratio

46. Sales in Units (thousands) Regular Price $.99 130 118 87 84 X 1 =104.75 X=119.58 Reduced Price $.89 145 143 120 131 X 2 =134.75 Cents-Off Coupon Regular Price 153 129 96 99 X 1 =119.25 Test Market A, B, or C Test Market D, E, or F Test Market G, H, or I Test Market J, K, or L Mean Grand Mean A Test Market Experiment on Pricing

47. ANOVA Summary Table Source of Variation Between groups Sum of squares –SSbetween Degrees of freedom –c-1 where c=number of groups Mean squared-MSbetween –SSbetween/c-1

48. ANOVA Summary Table Source of Variation Within groups Sum of squares –SSwithin Degrees of freedom –cn-c where c=number of groups, n= number of observations in a group Mean squared-MSwithin –SSwithin/cn-c

49. ANOVA Summary Table Source of Variation Total Sum of Squares –SStotal Degrees of Freedom –cn-1 where c=number of groups, n= number of observations in a group