{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "087P03NEND"
    }
  ],
  "body": "\n## Definition\n\nConsider a [[hypothesis test|087P03NEND]] of a null hypothesis $H_0 \\colon \\theta \\in \\Theta_0$ with a test statistic $T(X)$, where large values of $T$ are evidence against $H_0$. Let $t$ be the observed value of $T(X)$.\nThe p-value of the observation is\n$$p(t) = \\sup_{\\theta \\in \\Theta_0} P_\\theta\\big(T(X) \\ge t\\big).$$\nIf $H_0$ is simple, $\\Theta_0 = \\{\\theta_0\\}$, then $p(t) = P_{\\theta_0}(T(X) \\ge t)$.\n\n## Remarks\n\nFor a two-sided test with a statistic that is symmetric about $0$ under $H_0$, one uses $P_{\\theta_0}(|T(X)| \\ge |t|)$.\nThe test that rejects $H_0$ when $p \\le \\alpha$ has significance level $\\alpha$.\nIf $H_0$ is simple and $T(X)$ has a continuous law under $H_0$, then the random p-value $p(T(X))$ has the [[uniform distribution|0WGDPYR30Q]] on $[0, 1]$ under $H_0$.\nThe p-value is not the probability that $H_0$ is true.\n\n## Explanations\n\n### L1 human\n\nThe p-value tells how surprising your data would be if the null hypothesis were true.\nExample: a coin shows $14$ heads in $20$ tosses. If the coin is fair, the probability of at least $14$ heads is about $0.058$. This is the one-sided p-value.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0GGKJGZF6D",
    "kind": "definition",
    "msc": [
      "62F03"
    ],
    "name": "P-value",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/P-value"
      }
    ],
    "tag": "0EHYZFZ4PK",
    "uses": [
      "087P03NEND",
      "036756NFD6",
      "0WGDPYR30Q"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0EHYZFZ4PK/",
  "links": [
    {
      "field": "defines",
      "target": "0GGKJGZF6D"
    },
    {
      "field": "uses",
      "target": "087P03NEND"
    },
    {
      "field": "uses",
      "target": "036756NFD6"
    },
    {
      "field": "uses",
      "target": "0WGDPYR30Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0EHYZFZ4PK.md",
  "msc": [
    "62F03"
  ],
  "name": "P-value",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0EH/0EHYZFZ4PK.md",
  "status": null,
  "tag": "0EHYZFZ4PK",
  "uri": "m2://0EHYZFZ4PK",
  "url": "https://math2.tamnd.com/wiki/p-value-definition/",
  "writing": {
    "tag": "0EHYZFZ4PK",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}