---
tag: 087P03NEND
kind: concept
name: Statistical hypothesis test
aliases: [hypothesis-test, hypothesis-testing, significance-test, null-hypothesis]
msc: [62F03]
uses: [036756NFD6, 0EHYZFZ4PK, 0R41PVFA39]
sources:
  - url: https://en.wikipedia.org/wiki/Statistical_hypothesis_test
    locator: Definition of terms
  - url: https://en.wikipedia.org/wiki/Statistical_hypothesis_test
    locator: Performing a frequentist hypothesis test in practice
same_as:
  wikidata: Q210832
  wikipedia: Statistical hypothesis test
---

## Summary

A statistical hypothesis test is a rule that uses data to decide whether to reject a hypothesis about the law of the data.
One 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 $\Theta_1$.
One 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 $0.05$.

## Remarks

Equivalently, one can compute the [[p-value|0EHYZFZ4PK]] and reject $H_0$ when it is at most $\alpha$.
A 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.
The 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.
Not rejecting $H_0$ is not a proof that $H_0$ is true.

## Explanations

### L1 human

A hypothesis test asks whether the data are surprising under a default assumption, the null hypothesis.
If the data are too surprising, one rejects the assumption. The significance level fixes how often one accepts to reject a true assumption.