Mathematics 9709 · syllabus 6.5
Cambridge A Level Mathematics 9709 — Hypothesis tests topical past papers
Hypothesis tests on a binomial proportion, a Poisson mean and a normal population mean, with critical regions and Type I and Type II errors.
- Questions
- 277
- Years
- 2012–2026
- With mark scheme
- 100%
- Questions 2022–2026
- 90
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What Hypothesis tests covers
A hypothesis test asks whether data give enough evidence to reject a claim about a population. In Paper 6 (section 6.5) you state a null hypothesis H₀ and an alternative H₁ about a population parameter, such as p for a binomial, λ for a Poisson or μ for a mean, choose a one-tailed or two-tailed test from the wording, and compare a probability or a test statistic with the significance level. The conclusion must be stated in the context of the question.
Three settings are tested: a single observation from a binomial distribution and a single observation from a Poisson distribution, using exact probabilities or a normal approximation where it is appropriate; and a population mean, using the sample mean when the population is normal with known variance or the sample is large. You may also be asked for a critical region, the set of values that would lead you to reject H₀.
A Type I error is rejecting H₀ when it is true; a Type II error is not rejecting H₀ when it is false. You calculate their probabilities in specific cases: a Type I error has the probability of landing in the critical region when H₀ is true, and a Type II error needs a stated alternative value of the parameter.
In the syllabus
The headings this topic has in the Cambridge International 2026–2027 syllabus, as the board prints them.
- 6.5 Hypothesis tests
What you should be able to do
- State hypotheses about a population parameter, choosing one or two tails from the wording
- Carry out a test on a single binomial or Poisson observation using exact probabilities
- Use a normal approximation in a test where the conditions allow it
- Test a population mean using the distribution of the sample mean
- Find a critical region for a given significance level
- Calculate the probabilities of Type I and Type II errors
Key ideas
- H₀ states a single value, such as H₀: p = 0.3; H₁ is p < 0.3, p > 0.3 or p ≠ 0.3
- Reject H₀ if the probability of the observed value, or one more extreme, under H₀ is less than the significance level
- Test statistic for a mean: z = (x̄ − μ)/(σ/√n)
- Type I error: rejecting H₀ when it is true; P(Type I) = P(result in the critical region, given H₀ is true)
- Type II error: not rejecting H₀ when it is false; P(Type II) = P(result outside the critical region, given the stated alternative value)
How Hypothesis tests has been examined
From 2022 to 2026, Hypothesis tests was on Paper 6 · Probability & Statistics 2 in 14 of 14 exam series, worth about 37% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 6 · Probability & Statistics 2 | 90 | 37% | 14 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 6.
Filed in Momentum under: Hypothesis Testing using Binomial Distribution, Hypothesis Testing using Normal Distribution.
A worked example
A machine fills bottles with volumes that are normally distributed with standard deviation 4 ml, and the mean volume should be 500 ml. A random sample of 16 bottles has mean volume 498 ml. Test at the 5% significance level whether the mean volume is less than 500 ml.
- H₀: μ = 500, H₁: μ < 500, where μ ml is the population mean volume; a one-tailed test at the 5% level.
- Under H₀, X̄ ~ N(500, 4²/16), so the standard deviation of X̄ is 4/√16 = 1.
- Test statistic: z = (498 − 500)/1 = −2.
- P(Z < −2) = 1 − 0.9772 = 0.0228, which is less than 0.05; equivalently, −2 < −1.645.
- So reject H₀.
Answer: There is sufficient evidence at the 5% level that the mean volume is less than 500 ml.
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Writing hypotheses about the sample, such as H₀: x̄ = 50, instead of the population parameter μ.
- Finding P(X = 7) instead of P(X ≥ 7): the test uses the probability of the observed value or anything more extreme.
- Comparing each tail with the full significance level in a two-tailed test: each tail is compared with half of it.
- Concluding that H₀ is true, or that the claim is proved: the conclusion should say there is, or is not, sufficient evidence, in context.
- Giving the significance level as P(Type I error) in a binomial or Poisson test: the actual probability is that of the critical region, which is usually lower.
Exam technique
- Write the hypotheses with the parameter named, the significance level, and whether the test is one- or two-tailed before calculating anything.
- Show the probability or test statistic, the comparison with the significance level, and a conclusion in context, such as 'there is sufficient evidence at the 5% level that the mean time has increased'.
- For a critical region, give the probabilities on either side of the boundary to show where it lies.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
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Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.