Student's t-test

Student's t-test divides a normally distributed estimated effect by its estimated standard error. Under its null hypothesis the statistic has a Student's t-distribution, with degrees of freedom determined by the variance estimate.

For two independent samples from approximately normal distributions, Welch's t-test compares their means without assuming equal unknown variances. The displayed test statistic uses sample variances $s_X^2,s_Y^2$. Under the equal-mean null hypothesis, its reference Student t-distribution has approximate statistical degrees of freedomA p-value from this approximation addresses the specified statistical error model; it cannot distinguish a material effect from an unmeasured experimenter bias when sample and experimenter are confounded.

For independent normal samples of sizes $m,n$ with a common unknown variance and equal means under the null, $Z=(\overline{X}-\overline{Y})/(\sigma \sqrt{1/m+1/n})$ is standard normal. The independent centered sample sums of squares give $V=(m+n-2)S_p^2/{\sigma }^2\sim {\chi }_{m+n-2}^2$, independently of $Z$. Thus $Z/\sqrt{V/(m+n-2)}$ has the exact Student t-distribution, giving the statistic $(\overline{X}-\overline{Y})/(S_p\sqrt{1/m+1/n})$. A lower one-sided test uses its lower-tail probability.

For an independent sample from a normal distribution, maximization of the likelihood over the unknown variance gives the likelihood ratio for a fixed mean against an unrestricted mean as displayed, where $t=\sqrt{n}(\bar{X}-{\mu }_0)/S$ and $S^2={\sum }_i(X_i-\bar{X}{)}^2/(n-1)$. It decreases with $|t|$. Under the null, the normal sample mean is independent of its centered sum of squares, so $t$ has the exact Student t-distribution with $n-1$ degrees of freedom. The resulting two-sided test requires no asymptotic chi-squared approximation.

In a fixed full-rank normal linear model, the studentized estimator of one coefficient has a noncentral t-distribution with noncentrality equal to that coefficient divided by its standard deviation. The fitted and residual orthogonal projections of the Gaussian noise are independent. Therefore the statistical power of its exact Student's t-test does not depend on the other regression coefficients, with design and variance fixed.