Null and alternative hypotheses, one-tailed and two-tailed tests, the significance level, critical regions, and the binomial hypothesis test.

What this dot point is asking

WJEC wants you to set up a null hypothesis $H_0$ and an alternative hypothesis $H_1$, choose a one-tailed or two-tailed test, work at a stated significance level, find a critical region or compare a tail probability, and carry out a binomial hypothesis test with a clear conclusion in context. This is the climax of the AS statistics section and is examined every series.

The answer

Hypotheses

Significance level and critical region

The significance level $\alpha$ (commonly 5 per cent) is the probability of rejecting $H_0$ when it is actually true. The critical region is the set of values of $X$ so extreme that, if observed, you reject $H_0$.

Carrying out a binomial test

The method is the same every time: assume $H_0$, model with $X \sim B(n, p_0)$, find the probability of a result as extreme or more extreme than observed, and compare with $\alpha$.

Examples in context

Example 1. A two-tailed test. A spinner is claimed to be fair ($p = 0.5$ for red). In 30 spins it shows red 21 times. Testing $H_1\!: p \neq 0.5$ at 5 per cent, each tail has $2.5$ per cent. Here $P(X \ge 21) = 1 - P(X \le 20) \approx 0.0214 < 0.025$, so reject $H_0$: there is evidence the spinner is not fair. The two-tailed test guards against bias in either direction.

Example 2. A non-significant result. Testing the same manufacturer's claim with only $3$ faulty in $25$, $P(X \ge 3) = 1 - P(X \le 2) \approx 0.463$, far above $0.05$. We do not reject $H_0$: the data are consistent with a 10 per cent fault rate. Failing to reject is not the same as proving $H_0$ true.

Try this

Q1. State suitable hypotheses to test whether a die is biased towards sixes. [2 marks]

• Cue. $H_0\!: p = \tfrac{1}{6}$, $H_1\!: p > \tfrac{1}{6}$ (one-tailed).

Q2. For a two-tailed test at the 10 per cent level, how much probability is in each tail? [1 mark]

• Cue. $5$ per cent in each tail.

Q3. $X \sim B(20, 0.3)$ under $H_0$. Observed $X = 10$, testing $H_1\!: p > 0.3$. The tail probability is $P(X \ge 10) \approx 0.048$. State the conclusion at 5 per cent. [2 marks]

• Cue. $0.048 < 0.05$, so reject $H_0$: evidence that $p$ exceeds $0.3$.