It determines if there is a relationship between two categorical variables within a single population. It compares observed data to expected data assuming no relationship.

It determines if the distribution of a categorical variable is the same across multiple populations. It compares the distribution of data across these populations.

All expected counts must be at least 5. This ensures that the chi-square distribution is a good approximation for the test statistic's distribution.

A small p-value (typically less than 0.05) suggests that the observed data is unlikely if the null hypothesis were true, leading us to reject H0.

Compare the p-value to the significance level (alpha). If p-value ≤ alpha, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. State the conclusion in the context of the problem.

Independence: One sample, tests for association between two categorical variables. | Homogeneity: Multiple samples, tests if the distribution of a categorical variable is the same across populations.

Observed Frequency: Actual counts from the sample data. | Expected Frequency: Counts expected if the null hypothesis is true (no association).

Rejecting H0: There is statistically significant evidence for the alternative hypothesis. | Failing to Reject H0: There is not enough evidence to support the alternative hypothesis.

$\chi^2 = \sum \frac{(O - E)^2}{E}$

$E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$

$df = (\text{number of rows} - 1) \times (\text{number of columns} - 1)$