Setting up null and alternative hypotheses, the significance level, one-tailed and two-tailed tests, hypothesis tests for a binomial proportion and for a normal mean, critical regions, and interpreting the conclusion in context.

What this dot point is asking

AQA wants you to set up null and alternative hypotheses, choose a significance level, run one-tailed and two-tailed tests for a binomial proportion and for the mean of a normal distribution with known variance, find critical regions, and state a clear conclusion in context. This is a Paper 3 staple and rewards a disciplined layout: hypotheses, model, calculation, comparison, conclusion.

Setting up a test

Choose the direction from the wording. "Has the rate fallen" is one-tailed lower; "has the rate changed" is two-tailed. State $H_0$ and $H_1$ in terms of the population parameter, not the sample statistic.

The significance level and errors

The significance level $\alpha$, often $5$ percent or $1$ percent, is the threshold below which you reject the null hypothesis. It is precisely the probability of a Type I error: rejecting $H_0$ when it is actually true. For a two-tailed test you split $\alpha$ between the two tails, so each tail carries $\frac{\alpha}{2}$.

Binomial proportion test

For a test on a proportion, model the count $X$ as binomial under $H_0$ and compute the probability of a result as extreme as, or more extreme than, the one observed.

Critical regions

The critical region is the set of values of the test statistic that would lead you to reject $H_0$. For the binomial you find the smallest (or largest) count whose tail probability is within $\alpha$; for the normal mean you compare the standardised statistic with the critical z value. Stating the critical region in advance lets you simply check whether the observed value lies inside it.

Test for a normal mean

For a sample of size $n$ from a normal distribution with known standard deviation $\sigma$, the sample mean under $H_0$ satisfies $\bar{X} \sim N\left(\mu_0, \frac{\sigma^2}{n}\right)$.

A reliable five-step layout

Examiners award method marks for a clear structure, so use the same skeleton every time. First, define the parameter and state $H_0$ and $H_1$ in terms of it. Second, state the distribution of the test statistic under $H_0$ (the binomial $B(n, p_0)$, or the sample mean $N(\mu_0, \sigma^2 / n)$). Third, decide the significance level and whether the test is one- or two-tailed, halving the level across the tails if two-tailed. Fourth, compute either the probability of the observed result or more extreme, or the standardised statistic and the critical value. Fifth, compare and write a conclusion in the context of the original claim.

A subtle point is the difference between the two equivalent approaches. The probability (p-value) approach compares $P(\text{observed or more extreme})$ with $\alpha$; the critical-region approach finds the boundary in advance and checks whether the observation falls beyond it. Both must reach the same decision, and AQA accepts either, but mixing them (comparing a probability with a z value, say) loses marks. Choose one and apply it consistently.