Hypothesis testing for a population mean, the null and alternative hypotheses, one-tailed and two-tailed tests, the test statistic and its $p$ value, the comparison with a significance level, the decision and its interpretation, and the meaning of Type I and Type II errors

What this dot point is asking

VCAA wants you to carry out a hypothesis test for a population mean: state the null and alternative hypotheses, choose a one-tailed or two-tailed test, compute the test statistic and its $p$ value, compare with the significance level, and state the decision and its meaning. You also need to understand Type I and Type II errors.

Setting up the hypotheses

The null hypothesis $H_0: \mu = \mu_0$ states the value being tested, usually a status quo or claimed value. The alternative hypothesis $H_1$ states what we suspect instead:

• two-tailed: $H_1: \mu \ne \mu_0$, when a difference in either direction matters;
• one-tailed: $H_1: \mu > \mu_0$ or $H_1: \mu < \mu_0$, when only one direction is of interest.

The choice of tails is made before seeing the data, from the question's wording.

The test statistic and p value

Assuming $H_0$ is true, the sample mean is approximately normal with mean $\mu_0$ and standard error $\frac{\sigma}{\sqrt{n}}$. The standardised test statistic is

the number of standard errors between the observed sample mean and the hypothesised mean. The $p$ value is the probability, computed under $H_0$, of getting a result at least as extreme as the one observed:

• one-tailed: $p = P(Z > z)$ or $P(Z < z)$ in the direction of $H_1$;
• two-tailed: $p = 2\,P(Z > |z|)$, doubling to count both tails.

A small $p$ value means the observed data would be unlikely if $H_0$ were true, which is evidence against $H_0$.

The decision

Compare $p$ with the chosen significance level $\alpha$ (commonly $0.05$):

• if $p < \alpha$: reject $H_0$ in favour of $H_1$ (the result is statistically significant);
• if $p \ge \alpha$: do not reject $H_0$ (insufficient evidence).

State the conclusion in context, not just "reject" or "not reject".

Type I and Type II errors

Because the decision is based on a sample, it can be wrong:

• a Type I error is rejecting $H_0$ when it is actually true; its probability is $\alpha$;
• a Type II error is failing to reject $H_0$ when it is actually false.

Lowering $\alpha$ reduces the Type I error rate but raises the chance of a Type II error.

Examples in context

Example 1. For a two-tailed test with $z = 2.1$, $p = 2\,P(Z > 2.1) = 2(0.0179) = 0.0358 < 0.05$, so reject $H_0$.

Example 2. Setting $\alpha = 0.01$ instead of $0.05$ makes rejection harder, lowering the Type I error rate.

Try this

Q1. Write $H_0$ and a two-tailed $H_1$ for testing whether a mean equals $20$. [2 marks]

• Cue. $H_0: \mu = 20$, $H_1: \mu \ne 20$.

Q2. Compute the test statistic for $\bar{x} = 27$, $\mu_0 = 25$, $\sigma = 6$, $n = 36$. [2 marks]

• Cue. $z = \frac{27 - 25}{6/6} = \frac{2}{1} = 2$.

Q3. Define a Type I error. [1 mark]

• Cue. Rejecting $H_0$ when it is actually true.