Hypothesis Tests for a Population Mean (9709 PS2 §6.5) — Mathematics

Introduction

A hypothesis test for a population mean allows us to use sample data to decide whether there is sufficient statistical evidence to challenge a pre-existing belief about the mean $\mu$ of a population. In the 9709 exam, this arises in two settings:

The population is normally distributed with known variance $\sigma^2$ .
A large sample ( $n \geq 30$ is sufficient in practice) is used, so the Central Limit Theorem guarantees that the sample mean $\bar{X}$ is approximately normally distributed regardless of the shape of the population.

In both cases the test statistic is based on the distribution of the sample mean $\bar{X}$ , which you met in The Distribution of the Sample Mean. Mastering this topic is essential: hypothesis test questions regularly carry 7–10 marks on Paper 6.

Core Concept

Setting up the hypotheses

Every hypothesis test starts with two competing statements about the unknown population mean $\mu$ :

Null hypothesis $H_0$ : $\mu = \mu_0$ (the "status quo" value being tested).
Alternative hypothesis $H_1$ : states the direction of interest.

Type of test	$H_1$	Rejection region
Two-tailed	$\mu \neq \mu_0$	Both tails, significance level $\alpha/2$ each
One-tailed (upper)	$\mu > \mu_0$	Upper tail, significance level $\alpha$
One-tailed (lower)	$\mu < \mu_0$	Lower tail, significance level $\alpha$

The significance level $\alpha$ (commonly 5% or 1%) is always stated before examining the data.

The test statistic

Under $H_0$ , the sample mean satisfies:

\bar{X} \sim N\!\left(\mu_0,\, \frac{\sigma^2}{n}\right)

We standardise to obtain the $z$ -statistic:

Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}

Under $H_0$ this has the standard normal distribution $N(0,1)$ .

Two equivalent methods

Method 1 – Critical value: Compare the calculated $z$ with the critical value $z_c$ from the normal table at the chosen significance level. Reject $H_0$ if $|z| > z_c$ (two-tailed) or if $z$ falls in the appropriate tail.

Method 2 – $p$ -value: Calculate $p = P(Z \geq |z_{\text{calc}}|)$ (one-tailed) or $2P(Z \geq |z_{\text{calc}}|)$ (two-tailed). Reject $H_0$ if $p < \alpha$ .

Both methods must lead to the same conclusion; the 9709 mark scheme accepts either.

Large-sample case

When the population variance is unknown but the sample is large ( $n$ large), replace the unknown $\sigma^2$ with the sample variance $s^2$ (using the unbiased estimator $s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2$ ):

Z \approx \frac{\bar{X} - \mu_0}{s / \sqrt{n}}

The distribution is still treated as $N(0,1)$ (not $t$ ). This is the key distinction: the $t$ -distribution is not on the 9709 syllabus.

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