Testing for Independence in Contingency Tables

State the null and alternative hypotheses for a test of independence in a contingency table.

Explain how the Pearson chi-square statistic is constructed and determine its degrees of freedom.

Identify when the chi-square test assumptions are violated (e.g., small expected counts).

Explain the logic behind Fisher's exact test and when to prefer it over the chi-square test.

Explain the concept of a permutation test as a general approach for hypothesis testing.

Apply chi-square, Fisher's exact, and permutation-based tests to real data and report findings correctly.