Type I and Type II Errors in Hypothesis Testing (9709 Statistics 2) — Mathematics

Introduction

In any hypothesis test, a decision is made on the basis of sample evidence. Because we are reasoning under uncertainty, there is always a chance of making the wrong decision — even when the procedure is carried out perfectly. The syllabus objective for this section requires you to understand and work with the two specific ways in which a hypothesis test can go wrong: a Type I error and a Type II error.

These concepts appear regularly in 9709 Paper 7 questions — often asking you to calculate the exact probability of each type of error, or to interpret what each error means in context.

Core Concept

Recall from hypothesis testing that we choose a critical region: a set of values for the test statistic that lead us to reject $H_0$ . No matter how carefully the critical region is chosen, two kinds of incorrect conclusion are possible.

The Two Types of Error

Error	What happened	Consequence
Type I	$H_0$ is true, but we reject it	False positive — we conclude an effect exists when it does not
Type II	$H_0$ is false, but we fail to reject it	False negative — we miss a real effect

A simple way to remember this:

Type I = you convict an innocent person (reject a true $H_0$ ).
Type II = you acquit a guilty person (fail to reject a false $H_0$ ).

Probability of Each Error

P(\text{Type I error}) = P(\text{reject } H_0 \mid H_0 \text{ is true})

This is exactly the significance level $\alpha$ when the critical region is chosen to achieve that level exactly. In discrete distributions (Binomial, Poisson), the actual probability of a Type I error is often less than or equal to the nominal significance level, because the critical region is chosen to keep $P(\text{Type I})$ as close to (but not exceeding) $\alpha$ as possible.

P(\text{Type II error}) = P(\text{fail to reject } H_0 \mid H_0 \text{ is false})

To calculate this, you must be given (or assume) a specific alternative value of the parameter, say $p = p_1$ or $\lambda = \lambda_1$ . The test statistic is then evaluated under this alternative distribution.

The Trade-Off

Reducing $P(\text{Type I})$ (by making the critical region smaller) typically increases $P(\text{Type II})$ , and vice versa. The only way to reduce both simultaneously is to increase the sample size.

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