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

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

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

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.

A statistical test used to analyze categorical data to determine if there's a significant association between variables or if observed data fits an expected pattern.

There is no association between the variables (for independence) or the distributions are the same (for homogeneity).

There is an association between the variables (for independence) or the distributions are different (for homogeneity).

The actual count of data points falling into a specific category.

The count we would expect in a cell if the null hypothesis were true.

The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.