Student's t-test
= Student's t-test
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= Student t-test
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<Student's t-test> calibrates a standardized mean contrast using a <Student's t-distribution> under a specified sampling model. For independent $X_1,\ldots,X_n\sim N(\mu,\sigma^2)$ with $n\ge2$ and unknown positive <variance>, the one-sample statistic is
$$
T=\frac{\sqrt n(\bar X-\mu_0)}S,\qquad
S^2=\frac1{n-1}\sum_i(X_i-\bar X)^2.
$$
Under 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 $V=(n-1)S^2/\sigma^2\sim\chi^2_{n-1}$ 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.