Null and alternative hypotheses, one- and two-tailed tests, significance levels and critical regions, hypothesis tests for a binomial proportion, for a Normal mean, and for a correlation coefficient.

What this dot point is asking

OCR wants you to set up null and alternative hypotheses, choose a one- or two-tailed test, work at a stated significance level, find and use a critical region, carry out a hypothesis test for a binomial proportion, test the mean of a Normal distribution using the distribution of the sample mean, and test a product moment correlation coefficient against critical values, always concluding in context.

The answer

Hypotheses and significance

A hypothesis test weighs sample evidence against a claim. The null hypothesis $H_0$ is the default (the parameter equals a stated value); the alternative $H_1$ is what you test for (greater, less, or not equal). The significance level (often $5\%$ or $1\%$) is the probability of rejecting $H_0$ when it is actually true, that is the risk you accept of a false alarm.

One- and two-tailed tests

If $H_1$ has a direction ($p > 0.2$ or $\mu < 500$), the test is one-tailed and the whole significance level sits in one tail. If $H_1$ is "not equal", the test is two-tailed and the level is split between the two tails (so $2.5\%$ in each tail for a $5\%$ test).

The critical region

The critical region is the set of outcomes so extreme that you reject $H_0$. You either compare the probability of the observed result (or more extreme) with the significance level, or find the critical region first and check whether the observation falls in it.

The method, step by step

State the hypotheses and the model, identify the tail(s) and significance level, find the relevant probability or critical region, compare, and state the conclusion in the context of the question. The contextual conclusion is always worth a mark.

Examples in context

A binomial proportion test

A correlation test

For bivariate data you can test whether there is genuine correlation in the population. The hypotheses are $H_0: \rho = 0$ (no correlation) against a one- or two-tailed alternative, and you compare the sample correlation coefficient $r$ with a critical value from tables for the sample size and significance level.

Try this

Q1. For a two-tailed test at the $5\%$ level, what probability sits in each tail? [1 mark]

• Cue. $2.5\%$ in each tail.

Q2. A sample of $25$ from $N(\mu, 9)$ has mean $\bar{x}$. State the standard deviation of $\bar{X}$. [2 marks]

• Cue. $\dfrac{\sigma}{\sqrt{n}} = \dfrac{3}{\sqrt{25}} = \dfrac{3}{5} = 0.6$.