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  "body": "\n## Definition\n\nConsider a [[hypothesis test|087P03NEND]] of $H_0 \\colon \\theta \\in \\Theta_0$ against $H_1 \\colon \\theta \\in \\Theta_1$ that rejects $H_0$ when the data $X$ are in a rejection region $R$.\nA type I error occurs when $H_0$ is true and the test rejects $H_0$. Its probability at $\\theta \\in \\Theta_0$ is $P_\\theta(X \\in R)$.\nA type II error occurs when $H_0$ is false and the test does not reject $H_0$. Its probability at $\\theta \\in \\Theta_1$ is $\\beta(\\theta) = P_\\theta(X \\notin R)$.\n\n## Remarks\n\nThe size of the test is $\\alpha = \\sup_{\\theta \\in \\Theta_0} P_\\theta(X \\in R)$.\nThe power of the test at $\\theta \\in \\Theta_1$ is $1 - \\beta(\\theta)$.\nFor a fixed sample size, a smaller rejection region lowers the probability of a type I error. It raises the probability of a type II error.\nIn the frequentist practice, one fixes a bound $\\alpha$ on the type I error, and then chooses a test with a small type II error.\n\n## Explanations\n\n### L1 human\n\nA type I error is a false alarm: the test says \"effect\" when there is none. A type II error is a miss: the test says \"no effect\" when there is one.\nFor a medical test, a positive result for a healthy person is a type I error. A negative result for an ill person is a type II error.\n",
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