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  "body": "\n## Summary\n\nA Student's t-test is a [[hypothesis test|087P03NEND]] whose test statistic has [[Student's t-distribution|0NQ5798SB1]] under the null hypothesis. It mostly tests hypotheses about the means of normal populations with unknown variance.\nIn 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\n$$t = \\frac{\\bar{X}_n - \\mu_0}{S_n / \\sqrt{n}},$$\nwith the [[sample mean|0QNRBHWV35]] $\\bar{X}_n$ and the [[sample standard deviation|0JN9P4A944]] $S_n$. Under $H_0$, $t \\sim t_{n - 1}$.\n\n## Remarks\n\nThe 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.\nFor two samples with unequal variances, Welch's t-test uses an approximate number of degrees of freedom.\nThe paired t-test is the one-sample t-test applied to the differences of paired observations.\nFor large samples, the t-test gives almost the same results as a test with the normal law, because $t_\\nu$ approaches $\\mathcal{N}(0, 1)$.\n\n## Explanations\n\n### L1 human\n\nA t-test checks whether an average differs from a given value, or whether two averages differ, more than chance alone can explain.\nIt divides the difference by its estimated standard error, and compares the result with the t-distribution.\n",
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