Common Hypothesis Tests

We survey the most important parametric hypothesis tests, each designed for specific situations regarding the population distribution, sample size, and known/unknown parameters.

Tests for Means

Definition

The Student's $t$ -test is used for testing the mean when the variance is unknown:

One-sample: $H_0: \mu = \mu_0$ . Test statistic: $T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}} \sim t_{n-1}$ under $H_0$ (assuming normality).
Two-sample (equal variances): $H_0: \mu_1 = \mu_2$ . Test statistic: $T = \frac{\bar{X}_1 - \bar{X}_2}{S_p\sqrt{1/n_1 + 1/n_2}}$ where $S_p^2 = \frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1 + n_2 - 2}$ is the pooled variance. Under $H_0$ : $T \sim t_{n_1+n_2-2}$ .
Paired: $H_0: \mu_D = 0$ where $D_i = X_i - Y_i$ . Apply the one-sample $t$ -test to the differences.

ExampleOne-sample $t$-test

A manufacturer claims light bulbs last $\mu_0 = 1000$ hours. A sample of $n = 16$ gives $\bar{x} = 980$ , $s = 40$ . Test at $\alpha = 0.05$ : $T = \frac{980 - 1000}{40/\sqrt{16}} = \frac{-20}{10} = -2.0$ With $t_{15, 0.025} = 2.131$ , since $|-2.0| < 2.131$ , we fail to reject $H_0$ .

Tests for Variances and Proportions

Definition

Key tests for other parameters:

Chi-squared test for variance: $H_0: \sigma^2 = \sigma_0^2$ . Statistic: $\chi^2 = \frac{(n-1)S^2}{\sigma_0^2} \sim \chi^2_{n-1}$
F-test for equality of variances: $H_0: \sigma_1^2 = \sigma_2^2$ . Statistic: $F = S_1^2/S_2^2 \sim F_{n_1-1, n_2-1}$
Z-test for proportion: $H_0: p = p_0$ . Statistic: $Z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}} \sim N(0,1)$ approximately

Goodness of Fit

RemarkChi-squared goodness-of-fit test

Pearson's chi-squared test assesses whether observed frequencies match expected frequencies under a hypothesized distribution. The test statistic $\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$ has an approximate $\chi^2_{k-1-p}$ distribution, where $k$ is the number of categories and $p$ is the number of estimated parameters. Large values of $\chi^2$ indicate poor fit.