Chi-Squared and Hypothesis Testing

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

where $O$ = observed, $E$ = expected.

Tests whether data follows a particular distribution.

$H_0$: Data follows the expected distribution. $H_1$: It doesn't.

Degrees of freedom: $\nu = k - 1$ (where $k$ = number of categories).

Tests whether two categorical variables are independent.

$H_0$: Variables are independent. $H_1$: They are not independent.

Expected frequency: $E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}$.

Degrees of freedom: $\nu = (r-1)(c-1)$.

If $\chi^2 > \chi^2_{\text{critical}}$ (from tables at significance level), reject $H_0$.

Or: if $p$-value < significance level, reject $H_0$.

• All expected frequencies should be ≥ 5.
• Data should be frequencies (not percentages).