Type I and Type II Errors

Every test can go wrong in two ways: crying wolf (rejecting a true $H_0$ , Type I) or missing the wolf (keeping a false $H_0$ , Type II). Their probabilities come straight from the critical region.

Cambridge 9709 Paper 6Edexcel YMA01 WST02Edexcel 9MA0 Paper 3Edexcel 9FM0 OptionAQA 7357 Paper 3AQA 7367 Paper 3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Define Type I and Type II errors in context.
Find $P(\text{Type I})$ as the probability of the critical region when $H_0$ is true.
Find $P(\text{Type II})$ as the probability of the acceptance region for a given true parameter.
Explain the trade-off between the two errors.

Pause and think

A smoke alarm can fail in two ways. Which is worse: going off when there is no fire, or staying silent when there is one? How would you make one less likely, and what happens to the other?

Section 2 of 14

Core idea

Start hereTwo ways to be wrong

A smoke alarm can go off with no fire (false alarm) or stay silent during a fire (missed detection). Hypothesis tests have the same two mistakes, and making one rarer makes the other more common.

The big idea

Type I error: rejecting a true $H_0$ ; probability = significance level (actual level for discrete tests). Type II error: not rejecting a false $H_0$ ; probability depends on the true parameter.

1. Definitions

\begin{array}{c|cc}&H_0\text{ true}&H_0\text{ false}\\\hline\text{reject}&\text{Type I}&\checkmark\\\text{don't reject}&\checkmark&\text{Type II}\end{array}

H₀: μ = 0 versus the truth μ = 2.5 (critical value 1.645)

Shaded right of the line under the

H_0

curve: Type I error (

5\%

). Shaded left of the line under the true curve: Type II error,

P(Z<1.645-2.5)=0.196

Worked example $B(20,0.3)$ test, critical region $X\ge10$

P(Type I)? P(Type II) if actually $p=0.5$ ?

Goal: Two probabilities.
Strategy: Use the critical region under each truth.

$P(X\ge10\mid p=0.3)=0.048$
Type I: reject although $H_0$ true.
$P(X\le9\mid p=0.5)=0.412$
Type II: fail to reject although $p=0.5$ .

Answer

0.048

;

0.412

✓ CheckType I is just under the nominal 5% (discrete) ✓.

Worked exampleNormal test

$H_0:\mu=0$ , $\frac\sigma{\sqrt n}=1$ , reject if $z>1.645$ . True $\mu=2.5$ . P(Type II)?

Goal: Probability.
Strategy: $P(\bar X<1.645\mid\mu=2.5)$ .

$P(Z<1.645-2.5)=P(Z<-0.855)$
Shift to the true mean.
$=0.196$
Table.

Answer

0.196

✓ CheckPower

=1-0.196=0.804

✓.

Key takeawayTrade-off Lower significance level: fewer Type I, more Type II. Larger samples reduce both.

Pause and think

Can one test make both errors?

Summary

Type I: reject true $H_0$ .
Type II: keep false $H_0$ .

Type I error: Rejecting a true $H_0$ .
Type II error: Not rejecting a false $H_0$ .

HistoryNeyman and Pearson. Jerzy Neyman and Egon Pearson introduced the two types of error in papers from 1928 to 1933. Fisher's tests had only a null hypothesis; Neyman and Pearson added an explicit alternative so that "the error of missing a real effect" could be measured too. Fisher disliked the idea, and the two camps argued for decades.

Fun factThe boy who cried wolf. A memory aid: in Aesop's fable, the villagers first commit a Type I error (believing a wolf that isn't there), then a Type II error (ignoring the boy when the wolf really comes). First false alarm, then missed alarm: I then II.

Keep reading for free

Create a free account to read the rest of this lesson: key formulae, worked examples, explore the graph, common mistakes, exam tip, guided practice and more. Your answers are checked and your progress is saved.

Create free account Sign in