Chapter 22 Bivariate Statistical Analysis: Differences Between Two Variables © 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. ZIKMUND BABIN CARR GRIFFIN BUSINESS MARKET RESEARCH EIGHTH EDITION

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–2 LEARNING OUTCOMES 1.Recognize when a bivariate statistical test is appropriate 2.Calculate and interpret a χ 2 test for a contingency table 3.Calculate and interpret an independent samples t-test comparing two means 4.Understand the concept of analysis of variance (ANOVA) 5.Interpret an ANOVA table After studying this chapter, you should be able to

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–3 What Is the Appropriate Test of Difference? Test of DifferencesTest of Differences  An investigation of a hypothesis that two (or more) groups differ with respect to measures on a variable.  Behavior, characteristics, beliefs, opinions, emotions, or attitudes Bivariate Tests of DifferencesBivariate Tests of Differences  Involve only two variables: a variable that acts like a dependent variable and a variable that acts as a classification variable.  Differences in mean scores between groups or in comparing how two groups’ scores are distributed across possible response categories.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–4 EXHIBIT 22.1 Some Bivariate Hypotheses

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–5 Cross-Tabulation Tables: The χ 2 Test for Goodness-of-Fit Cross-Tabulation (Contingency) TableCross-Tabulation (Contingency) Table  A joint frequency distribution of observations on two more variables. χ 2 Distributionχ 2 Distribution  Provides a means for testing the statistical significance of a contingency table.  Involves comparing observed frequencies ( O i ) with expected frequencies ( E i ) in each cell of the table.  Captures the goodness- (or closeness-) of-fit of the observed distribution with the expected distribution.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–6 Chi-Square Test χ² = chi-square statistic 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

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–7 Degrees of Freedom (d.f.) d.f.=(R-1)(C-1)

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–8 Example: Papa John’s Restaurants Univariate Hypothesis: Papa John’s restaurants are more likely to be located in a stand-alone location or in a shopping center. Bivariate Hypothesis: Stand-alone locations are more likely to be profitable than are shopping center locations.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–9 Example: Papa John’s Restaurants (cont’d) In this example, χ 2 = with 1 d.f.In this example, χ 2 = with 1 d.f. From Table A.4, the critical value at the 0.05 level with 1 d.f. is 3.84.From Table A.4, the critical value at the 0.05 level with 1 d.f. is Thus, we are 95 percent confident that the observed values do not equal the expected values.Thus, we are 95 percent confident that the observed values do not equal the expected values. But are the deviations from the expected values in the hypothesized direction?But are the deviations from the expected values in the hypothesized direction?

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–10 χ 2 Test for Goodness-of-Fit Recap Testing the hypothesis involves two key steps: 1. Examine the statistical significance of the observed contingency table. 2. Examine whether the differences between the observed and expected values are consistent with the hypothesized prediction.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–11 The t-Test for Comparing Two Means Independent Samples t-TestIndependent Samples t-Test  A test for hypotheses stating that the mean scores for some interval- or ratio-scaled variable grouped based on some less-than-interval classificatory variable are not the same.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–12 The t-Test for Comparing Two Means (cont’d) Pooled Estimate of the Standard ErrorPooled Estimate of the Standard Error  An estimate of the standard error for a t-test of independent means that assumes the variances of both groups are equal.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–13 EXHIBIT 22.2 Independent Samples t -Test Results

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–14 EXHIBIT 22.3 SAS t -Test Output

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–15 Comparing Two Means (cont’d) Paired-Samples t-TestPaired-Samples t-Test  Compares the scores of two interval variables drawn from related populations.  Used when means need to be compared that are not from independent samples.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–16 EXHIBIT 22.4 Example Results for a Paired Samples t -Test

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–17 The Z -Test for Comparing Two Proportions Z -Test for Differences of ProportionsZ -Test for Differences of Proportions  Tests the hypothesis that proportions are significantly different for two independent samples or groups.  Requires a sample size greater than thirty.  The hypothesis is: H o : π 1 = π 2 may be restated as: H o : π 1 - π 2 = 0

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–18 The Z -Test for Comparing Two Proportions Z -Test statistic for differences in large random samples:Z -Test statistic for differences in large random samples: 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 2 S p 1 -p 2 =pooled estimate of the standard errors of differences of proportions

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–19 The Z -Test for Comparing Two Proportions To calculate the standard error of the differences in proportions:To calculate the standard error of the differences in proportions:

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–20 One-Way Analysis of Variance (ANOVA) Analysis of Variance (ANOVA)Analysis of Variance (ANOVA)  An analysis involving the investigation of the effects of one treatment variable on an interval-scaled dependent variable.  A hypothesis-testing technique to determine whether statistically significant differences in means occur between two or more groups.  A method of comparing variances to make inferences about the means.  The substantive hypothesis tested is:  At least one group mean is not equal to another group mean.

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–21 Partitioning Variance in ANOVA Total VariabilityTotal Variability  Grand Mean  The mean of a variable over all observations.  SST = Total of (observed value-grand mean) 2

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–22 Partitioning Variance in ANOVA Between-Groups VarianceBetween-Groups Variance  The sum of differences between the group mean and the grand mean summed over all groups for a given set of observations.  SSB = Total of n group (Group Mean − Grand Mean) 2 Within-Group Error or VarianceWithin-Group Error or Variance  The sum of the differences between observed values and the group mean for a given set of observations  Also known as total error variance.  SSE = Total of (Observed Mean − Group Mean) 2

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–23 The F -Test F -TestF -Test  Used to determine whether there is more variability in the scores of one sample than in the scores of another sample.  Variance components are used to compute F-ratios  SSE, SSB, SST

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–24 EXHIBIT 22.6 Interpreting ANOVA

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–25 EXHIBIT 22.1–1 SPSS Output

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–26 EXHIBIT 22A.1 A Test-Market Experiment on Pricing

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–27 EXHIBIT 22A.2 ANOVA Summary Table

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–28 EXHIBIT 22A.3 Pricing Experiment ANOVA Table

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–29 EXHIBIT 22B.1 ANOVA Table for Randomized Block Designs

© 2010 South-Western/Cengage Learning. All rights reserved. May not be scanned, copied or duplicated, or posted to a publically accessible website, in whole or in part.22–30 EXHIBIT 22B.2 ANOVA Table for Two-Factor Design