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  "body": "\n## Summary\n\nA statistical hypothesis test is a rule that uses data to decide whether to reject a hypothesis about the law of the data.\nOne fixes a null hypothesis $H_0$ and an alternative hypothesis $H_1$, for example $H_0 \\colon \\theta \\in \\Theta_0$ and $H_1 \\colon \\theta \\in \\Theta_1$ with disjoint sets $\\Theta_0$ and $\\Theta_1$.\nOne computes a test statistic $T(X)$ and rejects $H_0$ when $T(X)$ is in a critical region $C$. The region is chosen so that $P_\\theta(T(X) \\in C) \\le \\alpha$ for each $\\theta \\in \\Theta_0$, for a significance level $\\alpha$ such as $0.05$.\n\n## Remarks\n\nEquivalently, one can compute the [[p-value|0EHYZFZ4PK]] and reject $H_0$ when it is at most $\\alpha$.\nA test can make [[two types of errors|0R41PVFA39]]: reject a true $H_0$, or fail to reject a false $H_0$. The level $\\alpha$ bounds the probability of the first error.\nThe power of the test at $\\theta \\in \\Theta_1$ is $P_\\theta(T(X) \\in C)$, the probability to reject $H_0$ when it is false.\nNot rejecting $H_0$ is not a proof that $H_0$ is true.\n\n## Explanations\n\n### L1 human\n\nA hypothesis test asks whether the data are surprising under a default assumption, the null hypothesis.\nIf the data are too surprising, one rejects the assumption. The significance level fixes how often one accepts to reject a true assumption.\n",
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