---
tag: 0R41PVFA39
kind: definition
name: Type I and type II errors
msc: [62F03]
defines: 03N8C8ZR4B
uses: [087P03NEND]
sources:
  - url: https://en.wikipedia.org/wiki/Type_I_and_type_II_errors
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Type_I_and_type_II_errors
    locator: Error rate
---

## Definition

Consider 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 $R$.
A type I error occurs when $H_0$ is true and the test rejects $H_0$. Its probability at $\theta \in \Theta_0$ is $P_\theta(X \in R)$.
A 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 $\beta(\theta) = P_\theta(X \notin R)$.

## Remarks

The size of the test is $\alpha = \sup_{\theta \in \Theta_0} P_\theta(X \in R)$.
The power of the test at $\theta \in \Theta_1$ is $1 - \beta(\theta)$.
For 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.
In the frequentist practice, one fixes a bound $\alpha$ on the type I error, and then chooses a test with a small type II error.

## Explanations

### L1 human

A 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.
For 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.