The Chicago Female Fashion Study ^9 surveyed customers to...

The Chicago Female Fashion Study ${ }^9$ surveyed customers to determine characteristics of the "frequent" shoppers at different department stores in the Chicago area. Suppose you are a marketing manager at one of the department stores. You would like to know if a customer's shopping frequency and her age are related. Here are the data: (FIGURE CAN'T COPY), (FIGURE CAN'T COPY) a) Is this a test of homogeneity or independence? b) Write an appropriate hypothesis. c) Are the conditions for inference satisfied? d) The calculation yields $\chi^2=26.084, \mathrm{P}$-value $=0.002$. State your conclusion. e) Given the standardized residuals in the table, state a complete conclusion.

The Chicago Female Fashion Study ${ }^9$ surveyed customers to determine characteristics of the "frequent" shoppers at different department stores in the Chicago area. Suppose you are a marketing manager at one of the department stores. You would like to know if a customer's shopping frequency and her age are related. Here are the data: (FIGURE CAN'T COPY), (FIGURE CAN'T COPY) a) Is this a test of homogeneity or independence?
b) Write an appropriate hypothesis.
c) Are the conditions for inference satisfied?
d) The calculation yields $\chi^2=26.084, \mathrm{P}$-value $=0.002$. State your conclusion.
e) Given the standardized residuals in the table, state a complete conclusion.

Key Concepts

Chi-Square Test

This is a statistical test that compares the observed frequencies in categorical data to the frequencies expected if there were no association between the variables. It helps in determining whether differences between observed and expected counts are due to random chance or a real underlying relationship.

Test of Independence vs. Homogeneity

Understanding this distinction is key: the chi-square test of independence determines whether two categorical variables are associated within a single population, while the test of homogeneity compares the distribution of a categorical variable across different populations. Here, evaluating the relationship between shopping frequency and age involves testing for independence.

Hypotheses Formulation

In chi-square testing, the null hypothesis typically asserts that there is no association between the variables (or no difference in distributions), and the alternative hypothesis asserts that an association exists (or that the distributions differ). Crafting clear hypotheses is essential for guiding the analysis.

Conditions for Inference

Before conducting a chi-square analysis, several conditions must be met, such as the use of independently sampled data, sufficiently large expected cell counts (generally at least 5 per cell), and the proper categorization of data. Verifying these conditions helps validate the reliability of the test results.

Standardized Residuals

These are values that indicate how much each cell's observed frequency deviates from its expected frequency, measured in standard deviation units. They are useful for identifying which cells significantly contribute to the overall chi-square statistic, providing insight into the specific nature of the relationship between variables.

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