3.7StatisticsStretch

Hypothesis Testing with the Normal Distribution

A hypothesis test on a normal mean uses the distribution of the sample mean. Because the sample mean of a normal variable is itself normal with a smaller spread, we can test claims about μ.

26 min The Normal Distribution

Edexcel A Level Maths: 3.7 Hypothesis Testing with the Normal DistributionCurated by Kepler, not produced by us. Watch it before the notes below.

Open on YouTube

What you'll be able to do

Know the distribution of the sample mean
State hypotheses about μ
Standardise the sample mean
Reach a conclusion in context

Distribution of the sample mean

If $X \sim N(\mu, \sigma^2)$ , the mean of a sample of size $n$ is $\bar{X} \sim N\left(\mu, \dfrac{\sigma^2}{n}\right)$ — same mean, but variance divided by $n$ .

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

Distribution of the sample mean.

Carrying out the test

State $H_0: \mu = \mu_0$ and an alternative. Standardise the observed sample mean using $\dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$ and compare with the critical value (or find the probability) at the given significance level.

Example 1State the standard deviation of

\bar{X}

for

\sigma = 6

n = 9

\dfrac{\sigma}{\sqrt{n}} = \dfrac{6}{\sqrt{9}}

= 2

Answer

2

Tip — The standard deviation of the sample mean is σ/√n — it shrinks as n grows.

Formula recap

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

Sample mean distribution.

z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}

Test statistic.

Common mistakes to avoid

Using σ instead of σ/√n for the sample mean.

The sample mean’s standard deviation is σ/√n.

Forgetting to halve the significance level for a two-tailed test.

Split it between the two tails.

Key takeaways

Sample mean: X̄ ~ N(μ, σ²/n).
Test statistic z = (x̄ − μ₀)/(σ/√n).
Compare with the critical value and conclude in context.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The distribution of the sample mean is:
- $N(\mu, \sigma^2/n)$
- $N(\mu, \sigma^2)$
- $N(\mu, n\sigma^2)$
- $N(0,1)$
Show answer and explanation
Answer: $N(\mu, \sigma^2/n)$
Variance ÷ n.
The standard deviation of X̄ is:
- $\sigma/\sqrt{n}$
- $\sigma$
- $\sigma/n$
- $\sigma\sqrt{n}$
Show answer and explanation
Answer: $\sigma/\sqrt{n}$
Standard error.
For σ = 6, n = 9, the sd of X̄ is:
- $2$
- $6$
- $0.67$
- $18$
Show answer and explanation
Answer: $2$
6/√9.
The null hypothesis concerns:
- $\mu$
- $\bar{x}$
- $\sigma$
- $n$
Show answer and explanation
Answer: $\mu$
Population mean.
As n increases, the spread of X̄:
- decreases
- increases
- stays the same
- becomes zero
Show answer and explanation
Answer: decreases
σ/√n shrinks.

Next step

Check your understanding

Work through questions on Hypothesis Testing with the Normal Distribution. Use the feedback to decide what needs another look.

3 past-paper questions · 9 marks

Practise this topic

Related lessons

All 7 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.