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  "body": "\n## Definition\n\nLet $X$ be a sample whose law $P_\\theta$ depends on an unknown parameter $\\theta \\in \\Theta$. Let $\\tau \\colon \\Theta \\to \\mathbb{R}$ be a function and let $\\gamma \\in (0, 1)$.\nA confidence interval for $\\tau(\\theta)$ with confidence level $\\gamma$ is a random interval $[L(X), U(X)]$, where $L \\le U$ are [[statistics|036756NFD6]], such that\n$$P_\\theta\\big(L(X) \\le \\tau(\\theta) \\le U(X)\\big) = \\gamma \\quad \\text{for each } \\theta \\in \\Theta.$$\nIf only \"$\\ge \\gamma$\" holds for each $\\theta$, the interval is a conservative confidence interval.\n\n## Remarks\n\nExample: let $X_1, \\dots, X_n$ be i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ with $\\sigma$ known. Then $\\bar{X}_n \\pm 1.96\\, \\sigma / \\sqrt{n}$ is a confidence interval for $\\mu$ with level approximately $0.95$.\nIf $\\sigma$ is unknown, one replaces $\\sigma$ with the sample standard deviation and $1.96$ with a quantile of Student's t-distribution with $n - 1$ degrees of freedom.\nThe probability refers to the random interval, not to $\\theta$. After one computes the interval from data, it either contains $\\tau(\\theta)$ or it does not.\nThe confidence level is often written as $1 - \\alpha$, for example $95\\%$ for $\\alpha = 0.05$.\n\n## Explanations\n\n### L1 human\n\nA $95\\%$ confidence interval comes from a procedure that captures the true value in $95\\%$ of repeated experiments.\nIt does not mean that the true value has a $95\\%$ probability to be in one particular computed interval.\n",
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