 Student's t-test

Student's t-test calibrates a standardized mean contrast using a Student's t-distribution under a specified sampling model. For independent $X_1,\dots ,X_n\sim N(\mu ,{\sigma }^2)$ with $n\ge 2$ and unknown positive variance, the one-sample statistic isUnder the null hypothesis $\mu ={\mu }_0$, $T\sim t_{n-1}$ exactly.

The exact law uses two facts: $Z=\sqrt{n}(\bar{X}-{\mu }_0)/\sigma$ is standard normal, and $V=(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$ independently of $Z$. Hence $T=Z/\sqrt{V/(n-1)}$, the defining construction of the Student's t-distribution. A normally distributed numerator divided by an arbitrary estimated standard error need not have this law.

For a two-sided level-$\alpha$ test, reject when $|T|>t_{n-1,1-\alpha /2}$; a directional alternative uses the corresponding one-sided quantile. The paired t-test applies the one-sample construction to independent pairs' differences. The exact pooled two-sample t statistic uses independent normal samples with a common variance; Welch's t-test instead estimates unequal variances and uses an approximate degrees-of-freedom calibration. The selected model and statistical degrees of freedom determine which reference distribution is justified.