A-Level Maths hypothesis testing: binomial, Poisson and the normal mean

Where hypothesis testing sits in each specification

Cambridge Paper 6 can only be taken after Paper 5, on the route Papers 1, 3, 5 and 6.

The key ideas

The test is carried out assuming H0 is true. The p-value is the probability, under H0, of getting the observed result or one more extreme in the direction of H1. For a one-tailed test, compare it with the significance level, for example 5%. For a two-tailed test, compare each tail with half the significance level.

The critical region is the set of values of the test statistic that would lead to rejecting H0. For discrete distributions such as the binomial you cannot usually hit 5% exactly, so the actual significance level is the probability of the critical region, which is at most 5%.

A Type I error is rejecting H0 when it is true, and its probability is the actual significance level. A Type II error is failing to reject H0 when it is false. Conclusions are never 'H0 is true': you either have evidence to reject it or you do not.

Worked example 1 (binomial, one-tailed): a shop says 30% of customers use a voucher. In a sample of 20, only 2 do. Test at 5% whether the proportion is lower

1. Let X be the number using a voucher in 20. Under H0, X ~ B(20, 0.3).
2. H0: p = 0.3. H1: p < 0.3.
3. P(X ≤ 2) = P(0) + P(1) + P(2) = 0.0008 + 0.0068 + 0.0278 = 0.0355.
4. 0.0355 < 0.05, so the result is significant. Reject H0.
5. Conclusion in context: there is evidence at the 5% level that fewer than 30% of customers use a voucher.
6. The critical region is X ≤ 2, because P(X ≤ 3) = 0.1071 is above 5%. The actual significance level is 3.55%.

Worked example 2 (Poisson): a call centre receives 4.5 calls an hour on average. One hour it receives 9. Test at 5% whether the rate has increased

1. Under H0, X ~ Po(4.5). H0: λ = 4.5. H1: λ > 4.5.
2. P(X ≥ 9) = 1 - P(X ≤ 8) = 1 - 0.9597 = 0.0403.
3. 0.0403 < 0.05, so reject H0.
4. Conclusion: there is evidence at the 5% level that the mean number of calls per hour has increased.

Worked example 3 (Cambridge P6, normal mean): bags are meant to hold 50 g with σ = 6 g. A sample of 36 has mean 48.2 g. Test at 5% whether the mean has changed

1. H0: μ = 50. H1: μ ≠ 50 (two-tailed).
2. Under H0 the sample mean is N(50, 6^2 ÷ 36), so the standard error is 6 ÷ 6 = 1.
3. Test statistic z = (48.2 - 50) ÷ 1 = -1.8.
4. For a two-tailed test at 5% the critical values are ±1.96. Since -1.96 < -1.8, z is not in the critical region.
5. Do not reject H0. There is insufficient evidence at the 5% level that the mean mass has changed.

Common mistakes that cost marks

• Writing hypotheses in words or with the sample value instead of the population parameter, such as H0: X = 2.
• Using P(X = 2) instead of P(X ≤ 2).
• Forgetting to halve the significance level in each tail for a two-tailed test.
• Concluding 'H0 is true' or 'the claim is proved'.
• A conclusion with no context. The final mark usually needs the words of the question.
• Using P(X ≥ 9) = 1 - P(X ≤ 9) instead of 1 - P(X ≤ 8).

Exam technique and how a tutor helps

Use the same five-step structure every time: define the variable and its distribution under H0, state H0 and H1 with the parameter, calculate the probability or test statistic, compare with the significance level, and write a conclusion in context with a non-assertive phrase such as 'there is evidence that'. Examiners award the marks to these steps, so a correct structure earns most of the marks even if one probability is wrong.

Tutors find that hypothesis testing goes wrong in the language rather than the arithmetic. In one-to-one lessons the student writes full tests and the tutor marks them against the official mark scheme wording, especially the hypotheses and the conclusion, until the structure is automatic.

Self-check: can you do these?

• Write H0 and H1 to test whether a coin is biased. (Answer: H0: p = 0.5, H1: p ≠ 0.5)
• What is the probability of a Type I error in example 1? (Answer: 0.0355)
• For a two-tailed test at 10%, what probability is in each tail? (Answer: 5%)
• Why can a binomial test not have a significance level of exactly 5%? (Answer: X is discrete, so tail probabilities jump in steps)
• Which Cambridge paper must be taken before Paper 6? (Answer: Paper 5)