Explanation

In tests of independence for contingency table data, the appropriate nonparametric test statistic follows a chi-square distribution.

Key Points:

1. Contingency Table Tests: When analyzing categorical data arranged in contingency tables (cross-tabulations), we use the chi-square test of independence.
2. Test Statistic: The test statistic is calculated as: $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$ where $O_i$ are observed frequencies and $E_i$ are expected frequencies under the null hypothesis of independence.
3. Distribution: Under the null hypothesis, this test statistic follows a chi-square distribution with degrees of freedom equal to $(r-1)(c-1)$, where $r$ is the number of rows and $c$ is the number of columns in the contingency table.

Why Not Other Options:

• F-distributed (A): F-distribution is used in ANOVA tests and regression analysis, not for contingency table independence tests.
• Normally distributed (B): Normal distribution is used for parametric tests like z-tests, not for nonparametric contingency table tests.

Application:

This test is commonly used to determine whether there is a significant association between two categorical variables, such as testing whether gender is independent of voting preference, or whether product preference is independent of age group.