Exploring Marketing Research William G. Zikmund Chapter 22: Bivariate Statistics- Tests of Differences

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C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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 Common Bivariate Tests

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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 tst Common Bivariate Tests

C opyright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between two independent groups Nominal Chi-square test

C opyright © 2000 by Harcourt, Inc. All rights reserved. Differences Between Groups Contingency Tables Cross-Tabulation Chi-Square allows testing for significant differences between groups “Goodness of Fit”

C opyright © 2000 by Harcourt, Inc. All rights reserved. Chi-Square Test x² = chi-square statistics O i = observed frequency in the i th cell E i = expected frequency on the i th cell

C opyright © 2000 by Harcourt, Inc. All rights reserved. R i = total observed frequency in the i th row C j = total observed frequency in the j th column n = sample size Chi-Square Test

D.f.=(R-1)(C-1) C opyright © 2000 by Harcourt, Inc. All rights reserved. Degrees of Freedom

C opyright © 2000 by Harcourt, Inc. All rights reserved. Awareness of Tire Manufacturer’s Brand MenWomenTotal Aware Unaware

C opyright © 2000 by Harcourt, Inc. All rights reserved. Chi-Square Test: Differences Among Groups Example

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X 2 =3.84 with 1 d.f.

C opyright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between two independent groups Interval and ratio t-test or Z-test

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. Null Hypothesis About Mean Differences Between Groups

C opyright © 2000 by Harcourt, Inc. All rights reserved. t-Test for Difference of Means

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. t-Test for Difference of Means

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. Pooled estimate of the standard error

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. Degrees of Freedom d.f. = n - k where: –n = n 1 + n 2 –k = number of groups

C opyright © 2000 by Harcourt, Inc. All rights reserved. t-Test for Difference of Means -example

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

C opyright © 2000 by Harcourt, Inc. All rights reserved. Comparing Two Groups when Comparing Proportions Percentage Comparisons Sample Proportion - P Population Proportion -

C opyright © 2000 by Harcourt, Inc. All rights reserved. Differences Between Two Groups when Comparing Proportions The hypothesis is: H o :  1   may be restated as: H o :  1   

C opyright © 2000 by Harcourt, Inc. All rights reserved. A Z-Test for Differences of Proportions or

C opyright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for Differences of Proportions

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for Differences of Proportions - to calculate the standard error of the difference of proportions

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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 - to calculate the standard error of the difference of proportions

C opyright © 2000 by Harcourt, Inc. All rights reserved. Z-Test for Differences of Proportions - to calculate the standard error of the difference of proportions - weighted average

C opyright © 2000 by Harcourt, Inc. All rights reserved. A Z-Test for Differences of Proportions

C opyright © 2000 by Harcourt, Inc. All rights reserved. A Z-Test for Differences of Proportions

C opyright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between three or more independent groups Interval or ratio One-way ANOVA

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance Hypothesis when comparing three groups  1    

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - F-Ratio

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Sum of squares

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Sum of squares

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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)

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Sum of squares

C opyright © 2000 by Harcourt, Inc. All rights reserved. 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)

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Sum of squares

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Sum of squares = 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Mean Squares Between

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Mean Square Within

C opyright © 2000 by Harcourt, Inc. All rights reserved. Analysis of Variance - Calculation of F-Ratio

C opyright © 2000 by Harcourt, Inc. All rights reserved. A Test Market Experiment on Pricing Sales in Units (thousands) Regular Price $ X 1 = X= Reduced Price $ X 2 = Cents-Off Coupon Regular Price X 1 = 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

C opyright © 2000 by Harcourt, Inc. All rights reserved. ANOVA Summary Table Source of Variation Between Groups Sum of Squares –SS BETWEEN Degrees of Freedom –c-1 where c=number of groups Mean Squared-MS BETWEEN –SS BETWEEN /c-1

C opyright © 2000 by Harcourt, Inc. All rights reserved. Within Groups Sum of Squares –SS WITHIN Degrees of Freedom –cn-c where c=number of groups, n= number of observations in a group Mean Squared-MS WITHIN –SS WITHIN /cn-c ANOVA Summary Table Source of Variation

C opyright © 2000 by Harcourt, Inc. All rights reserved. Total Sum of Squares –SS TOTAL Degrees of Freedom –cn-1 where c=number of groups, n= number of observations in a group ANOVA Summary Table Source of Variation

C opyright © 2000 by Harcourt, Inc. All rights reserved. Type of Measurement Differences between two independent groups Ordinal Mann-Whitney U-test Wilcocon test