chapter 22 bivariate statistical analysis: differences between two variables © 2010...

Chapter 22Chapter 22Bivariate Statistical Bivariate Statistical

Analysis: Analysis: Differences Differences

Between Two Between Two VariablesVariables

© 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.

© 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 BABINCARR 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 OUTCOMESLEARNING OUTCOMESLEARNING OUTCOMESLEARNING OUTCOMES

1.1. Recognize when a bivariate statistical test is Recognize when a bivariate statistical test is appropriateappropriate

2.2. Calculate and interpret a Calculate and interpret a χχ22 test for a contingency table test for a contingency table

3.3. Calculate and interpret an independent samples Calculate and interpret an independent samples tt-test -test comparing two meanscomparing two means

4.4. Understand the concept of analysis of variance Understand the concept of analysis of variance (ANOVA)(ANOVA)

5.5. Interpret an ANOVA tableInterpret an ANOVA table

After studying this chapter, you should be able to


What Is the Appropriate Test of What Is the Appropriate Test of Difference?Difference?• Test of DifferencesTest of Differences

An investigation of a hypothesis that two (or more) An investigation of a hypothesis that two (or more) groups differ with respect to measures on a variable.groups differ with respect to measures on a variable. Behavior, characteristics, beliefs, opinions, emotions, or Behavior, characteristics, beliefs, opinions, emotions, or

attitudesattitudes

• Bivariate Tests of DifferencesBivariate Tests of Differences Involve only two variables: a variable that acts like a Involve only two variables: a variable that acts like a

dependent variable and a variable that acts as a dependent variable and a variable that acts as a classification variable.classification variable. Differences in mean scores between groups or in comparing Differences in mean scores between groups or in comparing

how two groups’ scores are distributed across possible how two groups’ scores are distributed across possible response categories.response categories.


EXHIBIT 22.EXHIBIT 22.11 Some Bivariate HypothesesSome Bivariate Hypotheses


Cross-Tabulation Tables: The Cross-Tabulation Tables: The χχ22 Test Test for Goodness-of-Fitfor Goodness-of-Fit• Cross-Tabulation (Contingency) TableCross-Tabulation (Contingency) Table

A joint frequency distribution of observations on two A joint frequency distribution of observations on two more variables.more variables.

• χχ22 Distribution Distribution Provides a means for testing the statistical Provides a means for testing the statistical

significance of a contingency table.significance of a contingency table. Involves comparing observed frequencies (Involves comparing observed frequencies (OOii) with ) with

expected frequencies (expected frequencies (EEii) in each cell of the table.) in each cell of the table.

Captures the goodness- (or closeness-) of-fit of the Captures the goodness- (or closeness-) of-fit of the observed distribution with the expected distribution.observed distribution with the expected distribution.


Chi-Square TestChi-Square Test

i

ii

E

)²E(O χ²

χ² = chi-square statisticOi = observed frequency in the ith cellEi = expected frequency on the ith cell

n

CRE ji

ij

Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size


Degrees of Freedom (d.f.)Degrees of Freedom (d.f.)

d.f.=(R-1)(C-1)d.f.=(R-1)(C-1)


Example: Papa John’s RestaurantsExample: Papa John’s RestaurantsUnivariate 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.


Example: Papa John’s Restaurants Example: Papa John’s Restaurants (cont’d)(cont’d)• In this example, In this example, χχ2 2 = 22.16 with 1 d.f. = 22.16 with 1 d.f.• From Table A.4, the critical value at the 0.05 From Table A.4, the critical value at the 0.05

level with 1 d.f. is 3.84.level with 1 d.f. is 3.84.• Thus, we are 95 percent confident that the Thus, we are 95 percent confident that the

observed values do not equal the expected observed values do not equal the expected values.values.

• But are the deviations from the expected values But are the deviations from the expected values in the hypothesized direction?in the hypothesized direction?


χχ22 Test for Goodness-of-Fit Recap Test for Goodness-of-Fit Recap

Testing the hypothesis involves two key steps:Testing the hypothesis involves two key steps:1.1. Examine the statistical significance of the observed Examine the statistical significance of the observed

contingency table.contingency table.

2.2. Examine whether the differences between the Examine whether the differences between the observed and expected values are consistent with observed and expected values are consistent with the hypothesized prediction.the hypothesized prediction.


The The tt-Test for Comparing Two Means-Test for Comparing Two Means

• Independent Samples Independent Samples tt-Test-Test A test for hypotheses stating that the mean scores for A test for hypotheses stating that the mean scores for

some interval- or ratio-scaled variable grouped based some interval- or ratio-scaled variable grouped based on some less-than-interval classificatory variable are on some less-than-interval classificatory variable are not the same.not the same.

means random ofy Variabilit

2 MeanSample - 1 MeanSample t

21

21 XXS

t


The The tt-Test for Comparing Two Means -Test for Comparing Two Means (cont’d)(cont’d)

• Pooled Estimate of the Standard ErrorPooled Estimate of the Standard Error An estimate of the standard error for a An estimate of the standard error for a tt-test of -test of

independent means that assumes the variances of independent means that assumes the variances of both groups are equal.both groups are equal.

2121

222

211 11

2

1121 nnnn

SnSnS XX

))(


EXHIBIT 22.EXHIBIT 22.22 Independent Samples Independent Samples tt-Test Results-Test Results


EXHIBIT 22.EXHIBIT 22.33 SAS SAS tt-Test Output-Test Output


Comparing Two Means (cont’d)Comparing Two Means (cont’d)

• Paired-SamplesPaired-Samples t t-Test-Test Compares the scores of two interval variables drawn Compares the scores of two interval variables drawn

from related populations.from related populations. Used when means need to be compared that are not Used when means need to be compared that are not

from independent samples.from independent samples.


EXHIBIT 22.EXHIBIT 22.44 Example Results for a Paired Samples Example Results for a Paired Samples tt-Test-Test


The The ZZ-Test for Comparing Two -Test for Comparing Two ProportionsProportions• ZZ-Test for Differences of Proportions-Test for Differences of Proportions

Tests the hypothesis that proportions are significantly Tests the hypothesis that proportions are significantly different for two independent samples or groups.different for two independent samples or groups.

Requires a sample size greater than thirty.Requires a sample size greater than thirty.

The hypothesis is:The hypothesis is: HHoo: : ππ11 = = ππ22

may be restated as:may be restated as: HHoo: : ππ11 - - ππ22 = = 0 0


The The ZZ-Test for Comparing Two -Test for Comparing Two ProportionsProportions• ZZ-Test statistic for differences in large random -Test statistic for differences in large random

samples:samples:

21

2121

ppS

ppZ

p1 = sample portion of successes in Group 1

p2 = sample portion of successes in Group 2

1 1) = hypothesized population proportion 1

minus hypothesized population proportion 2

Sp1-p2 = pooled estimate of the standard errors of

differences of proportions


The The ZZ-Test for Comparing Two -Test for Comparing Two ProportionsProportions• To calculate the standard error of the differences To calculate the standard error of the differences

in proportions:in proportions:

21

1121 nn

qpS pp


One-Way Analysis of Variance One-Way Analysis of Variance (ANOVA)(ANOVA)• Analysis of Variance (ANOVA)Analysis of Variance (ANOVA)

An analysis involving the investigation of the effects of An analysis involving the investigation of the effects of one treatment variable on an interval-scaled one treatment variable on an interval-scaled dependent variable.dependent variable.

A hypothesis-testing technique to determine whether A hypothesis-testing technique to determine whether statistically significant differences in means occur statistically significant differences in means occur between two or more groups.between two or more groups.

A method of comparing variances to make inferences A method of comparing variances to make inferences about the means.about the means.

The substantive hypothesis tested is:The substantive hypothesis tested is: At least one group mean is not equal to another group mean.At least one group mean is not equal to another group mean.


Partitioning Variance in ANOVAPartitioning Variance in ANOVA

• Total VariabilityTotal Variability Grand MeanGrand Mean

The mean of a variable over all observations.The mean of a variable over all observations.

SST = Total of (observed value-grand mean)SST = Total of (observed value-grand mean)22


Partitioning Variance in ANOVAPartitioning Variance in ANOVA

• Between-Groups VarianceBetween-Groups Variance The sum of differences between the group mean and The sum of differences between the group mean and

the grand mean summed over all groups for a given the grand mean summed over all groups for a given set of observations.set of observations.

SSB = Total of SSB = Total of nngroupgroup(Group Mean − Grand Mean)(Group Mean − Grand Mean)22

• Within-Group Error or VarianceWithin-Group Error or Variance

The sum of the differences between observed values The sum of the differences between observed values and the group mean for a given set of observationsand the group mean for a given set of observations Also known as total error variance.Also known as total error variance.

SSE = Total of (Observed Mean − Group Mean)SSE = Total of (Observed Mean − Group Mean)22


The The FF-Test-Test

• FF-Test-Test Used to determine whether there is more variability in Used to determine whether there is more variability in

the scores of one sample than in the scores of the scores of one sample than in the scores of another sample.another sample.

Variance components are used to compute Variance components are used to compute FF-ratios-ratios SSE, SSB, SSTSSE, SSB, SST

groupswithinVariance

groupsbetweenVarianceF


EXHIBIT 22.EXHIBIT 22.66 Interpreting ANOVAInterpreting ANOVA


EXHIBIT 22.1EXHIBIT 22.1–1–1

SPSS OutputSPSS Output


EXHIBIT 22A.1EXHIBIT 22A.1 A Test-Market Experiment on PricingA Test-Market Experiment on Pricing


EXHIBIT 22A.2EXHIBIT 22A.2 ANOVA Summary TableANOVA Summary Table


EXHIBIT 22A.3EXHIBIT 22A.3 Pricing Experiment ANOVA TablePricing Experiment ANOVA Table


EXHIBIT 22B.1EXHIBIT 22B.1 ANOVA Table for Randomized Block DesignsANOVA Table for Randomized Block Designs


EXHIBIT 22B.2EXHIBIT 22B.2 ANOVA Table for Two-Factor DesignANOVA Table for Two-Factor Design

chapter 22 bivariate statistical analysis: differences between two variables © 2010...

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