Hypothesis tests for the mean of a Poisson distribution, tests for a population mean using the normal distribution, one-tailed and two-tailed tests, and the meaning of Type I and Type II errors.

What this dot point is asking

AQA wants you to set up and carry out hypothesis tests for the mean of a Poisson distribution and for a population mean using the normal distribution, to distinguish one-tailed and two-tailed tests, to compare a test statistic with a critical value or a probability with the significance level, and to explain Type I and Type II errors.

Setting up a test

Every test follows the same skeleton, and marks are lost by skipping a step rather than by hard calculation. State the null hypothesis $H_0$ (the value being assumed) and the alternative $H_1$ (what you are testing for). Choose the significance level, usually $5\%$ or $1\%$. Decide on one tail or two: a claim that a parameter has increased or decreased is one-tailed, while a claim that it has merely changed is two-tailed, splitting the significance level between the two tails. Then either find the critical region in advance and check whether the observation falls in it, or compute the probability of a result as extreme as observed and compare it with the significance level. Finally, state the conclusion in the context of the original problem, not just as accept or reject.

Test for a Poisson mean

For a Poisson test the test statistic is the observed count itself, and you work with exact Poisson probabilities or cumulative tables rather than a continuous approximation. For an upper-tail test the relevant probability is $P(X \geq \text{observed})$, computed as $1 - P(X \leq \text{observed} - 1)$ from cumulative tables; for a lower-tail test it is $P(X \leq \text{observed})$ directly.

Test for a population mean

This $Z$ test relies on the sampling distribution of the mean. If the population is normal with known variance, the sample mean $\bar{X}$ is normally distributed with mean $\mu$ and standard error $\frac{\sigma}{\sqrt{n}}$, so standardising gives the $Z$ statistic. By the central limit theorem the same test is approximately valid for a large sample from any population, even one that is not normal, which is why it appears so widely. A large absolute value of $Z$ means the observed sample mean is many standard errors away from the hypothesised mean, which is unlikely if $H_0$ is true, so $H_0$ is rejected.

Type I and Type II errors

The two error types pull against each other, which is the heart of choosing a significance level. A very small significance level makes you reluctant to reject $H_0$, so you rarely raise a false alarm (low Type I rate) but more often miss a genuine effect (high Type II rate). Increasing the sample size is the only way to reduce both at once, because a larger sample sharpens the sampling distribution and separates the competing hypotheses more cleanly. In context, a Type I error is a false positive (acting on a change that is not real) and a Type II error is a false negative (missing a change that is real); which is worse depends on the practical consequences.