Introduction to Hypothesis Testing (9709 PS2 §6.5) — Mathematics

Introduction

Hypothesis testing is the formal statistical procedure by which we use sample data to decide whether there is sufficient evidence to challenge a previously held belief about a population parameter. In the 9709 exam, hypothesis tests appear regularly in Paper 6 (Probability & Statistics 2) and require you to: state hypotheses correctly, identify the critical region, compute a test statistic, and — crucially — interpret the outcome in plain English within the context given. Losing marks through vague or missing conclusions is one of the most common avoidable errors.

Core Concept

The Big Picture

We begin with an assumption about a population parameter (e.g. the probability $p$ of success, or the mean $\mu$ of a distribution). We collect a sample and ask: if the assumption were true, how surprising would this sample result be? If it is surprising enough, we reject the assumption.

Null and Alternative Hypotheses

Term	Symbol	Meaning
Null hypothesis	$H_0$	The default claim about the parameter — always contains an equality
Alternative hypothesis	$H_1$	What we are trying to find evidence for — contains $\neq$ , $<$ , or $>$

$H_0$ is the hypothesis that is tested. We never "prove" $H_0$ ; we either reject it or fail to reject it.

Significance Level

The significance level $\alpha$ (typically $1\%$ , $5\%$ , or $10\%$ ) is the probability threshold below which a result is considered sufficiently unlikely under $H_0$ to justify rejection. It is set before the test is carried out.

\alpha = P(\text{rejecting } H_0 \mid H_0 \text{ is true})

This is the probability of a Type I error (rejecting a true $H_0$ ).

Test Statistic

The test statistic is a single numerical value calculated from the sample data. It is compared against the critical region to make a decision. For example, if $X \sim B(n, p)$ under $H_0$ , the observed value of $X$ is the test statistic.

Critical (Rejection) Region and Acceptance Region

The critical region (also called the rejection region) is the set of values of the test statistic for which $H_0$ is rejected. Its boundary value(s) are called critical value(s).

The acceptance region is the complementary set — values for which we do not reject $H_0$ .

For a test at significance level $\alpha$ :

P(\text{test statistic falls in critical region} \mid H_0) \leq \alpha

In discrete distributions (e.g. binomial, Poisson) the actual probability of the critical region is usually less than $\alpha$ , because we cannot achieve exactly $\alpha$ with integer boundaries.

One-Tailed vs Two-Tailed Tests

Feature	One-tailed test	Two-tailed test
$H_1$ form	$p < p_0$ or $p > p_0$	$p \neq p_0$
Critical region	One tail only	Split between both tails
Each tail's significance	Full $\alpha$	$\alpha/2$ per tail
Use when…	Prior reason to suspect direction of change	No prior reason; any change matters

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