Student's t-test

Summary

A Student's t-test is a hypothesis test whose test statistic has Student's t-distribution under the null hypothesis. It mostly tests hypotheses about the means of normal populations with unknown variance. In the one-sample t-test, $X_1, \dots, X_n$ are i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with unknown $\sigma$, and $H_0 \colon \mu = \mu_0$. The statistic is $$t = \frac{\bar{X}_n - \mu_0}{S_n / \sqrt{n}},$$ with the sample mean $\bar{X}_n$ and the sample standard deviation $S_n$. Under $H_0$, $t \sim t_{n - 1}$.

Remarks

The two-sample t-test compares the means of two independent normal samples with sizes $n_1$, $n_2$ and equal variances. It uses the pooled standard deviation $s_p$ and $t = (\bar{X}_1 - \bar{X}_2) / \big(s_p \sqrt{1/n_1 + 1/n_2}\big)$, with $n_1 + n_2 - 2$ degrees of freedom. For two samples with unequal variances, Welch's t-test uses an approximate number of degrees of freedom. The paired t-test is the one-sample t-test applied to the differences of paired observations. For large samples, the t-test gives almost the same results as a test with the normal law, because $t_\nu$ approaches $\mathcal{N}(0, 1)$.

Explanations

L1 human

A t-test checks whether an average differs from a given value, or whether two averages differ, more than chance alone can explain. It divides the difference by its estimated standard error, and compares the result with the t-distribution.