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 Video by Zeeshan Zamurred The Normal Distribution

Edexcel A Level Maths: 3.7 Hypothesis Testing with the Normal DistributionWatch the full walkthrough before the notes below.

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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 $X \sim N (μ, σ^{2})$ , the mean of a sample of size $n$ is $\overset{ˉ}{X} \sim N (μ, \frac{σ ^{2}}{n})$ — same mean, but variance divided by $n$ .

\overset{ˉ}{X} \sim N (μ, \frac{σ ^{2}}{n})

Distribution of the sample mean.

Carrying out the test

State $H_{0} : μ = μ_{0}$ and an alternative. Standardise the observed sample mean using $\frac{x ˉ - μ _{0}}{σ / n}$ and compare with the critical value (or find the probability) at the given significance level.

\frac{σ}{n} = \frac{6}{9}

= 2

Answer

2

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

Formula recap

\overset{ˉ}{X} \sim N (μ, \frac{σ ^{2}}{n})

Sample mean distribution.

z = \frac{x ˉ - μ _{0}}{σ / 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

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