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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.
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.
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.
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.
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
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
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
• 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