AQA A-Level Mathematics: O: Statistical hypothesis testing (7357) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover Section O: Statistical hypothesis testing (O1 to O3) of the AQA A-level Mathematics (7357) specification, version 1.3, for A-level exams from June 2018 onwards. Section O is listed under Paper 3. A calculator is required in every 7357 paper, so every question here assumes you have one with binomial, Normal and inverse Normal functions. Give probabilities to 3 significant figures unless a question says otherwise.

Before you start, see the hypothesis testing study guide or the revision notes. The 7357 course hub and printable checklist list every topic, and the free 10-minute diagnostics help you find gaps.

Questions

1. In the context of a hypothesis test:

(a) State what is meant by the critical region. [1] (b) State what is meant by a p-value. [1]

2. A games designer states that a reward box contains a rare item with probability 0.08. A player suspects that the probability is lower than this. Define a suitable parameter and write down the hypotheses for a test of the player’s suspicion. [2]

3. In a town, 40% of adults used to cycle to work. After a new car park opens, a random sample of 30 adults finds that 7 cycle to work. Test at the 5% significance level whether the proportion who cycle to work has decreased. [5]

4. A seed supplier states that 15% of its seeds fail to germinate. A gardener plants a random sample of 20 seeds and suspects that the proportion failing is higher than stated.

(a) Using a 5% significance level, find the critical region for the number of seeds that fail to germinate. [3] (b) State the probability that the test incorrectly rejects the null hypothesis. [1]

5. A biased coin is thought to land heads with probability 0.3. It is thrown 50 times to test whether this probability is correct, using a 5% significance level.

(a) Find the critical region for X, the number of heads. [4] (b) Find the actual significance level of the test. [1] (c) The coin lands heads 21 times. State the conclusion of the test in context. [1]

6. For a random sample of 15 cars, the product moment correlation coefficient between age and resale value is r = −0.47. The critical value for a 1-tail test at the 5% level with a sample of 15 is 0.4409.

(a) Write down hypotheses to test for negative correlation in the population. [1] (b) Carry out the test at the 5% level. [2] (c) Describe what the value r = −0.47 shows about the scatter diagram of the data. [1]

7. For 40 randomly chosen days, the correlation coefficient between daily rainfall and the number of umbrellas sold in a shop is r = 0.29. For a test of H₀: ρ = 0 against H₁: ρ ≠ 0, the p-value is 0.069.

(a) Carry out the test at the 5% significance level. [2] (b) State, with a reason, whether the conclusion would be different at the 10% level. [1]

8. A phone battery gives a video playback time that is Normally distributed with mean 250 minutes and standard deviation 12 minutes. After a software update, a random sample of 20 phones has a mean playback time of 244.8 minutes. Assuming the standard deviation is unchanged, test at the 5% level whether the mean playback time has decreased. [6]

9. The lengths of a species of lizard are Normally distributed with standard deviation 2.4 cm. A field guide states the mean length is 18.5 cm. A random sample of 36 lizards from one island is measured to test whether the mean length on the island is different from 18.5 cm.

(a) Find the critical region for the sample mean X̄ at the 5% significance level. [4] (b) The sample mean is 19.4 cm. State, with a reason, the conclusion of the test in context. [2]

10. A website records that 12% of visitors click on a particular advert. After a redesign, 9 out of a random sample of 40 visitors click on the advert.

(a) Test at the 5% significance level whether the proportion who click has increased. [5] (b) Find the critical region for this test at the 1% significance level. [2] (c) Explain what the 1% significance level means in the context of this test. [1]

11. A machine makes metal pins whose lengths are Normally distributed with mean 4.2 cm and standard deviation 0.5 cm. An engineer suspects the mean length has increased. A random sample of 25 pins has mean length 4.37 cm.

(a) Assuming the standard deviation is unchanged, test the engineer’s suspicion at the 2.5% significance level. [5] (b) The same sample mean, 4.37 cm, is obtained from a larger random sample of size n. Find the smallest value of n for which the result would be significant at the 2.5% level. [3]

Answers

1. (a) The set of values of the test statistic that lead to the null hypothesis being rejected [1]. (b) The probability, assuming H₀ is true, of obtaining a result at least as extreme as the one observed [1]. [2] Examiner insight: the p-value definition needs “assuming H₀ is true” and “at least as extreme”.

2. Let p be the probability that a reward box contains the rare item [1]. H₀: p = 0.08, H₁: p < 0.08 [1]. [2] Examiner insight: an undefined letter in the hypotheses loses the definition mark.

3. Let p = proportion of all adults in the town who cycle to work. H₀: p = 0.4, H₁: p < 0.4 [1]. Under H₀, X ~ B(30, 0.4) [1]. p-value = P(X ≤ 7) [1] = 0.0435 [1]. 0.0435 < 0.05, so reject H₀. There is sufficient evidence at the 5% level to suggest that the proportion of adults who cycle to work has decreased [1]. [5] Examiner insight: the final mark needs a comparison with 0.05 and a non-definite conclusion in context.

4. (a) Under H₀, X ~ B(20, 0.15), H₁: p > 0.15 [1]. P(X ≥ 7) = 0.0219 and P(X ≥ 6) = 0.0673 [1]. Critical region X ≥ 7 [1]. (b) P(X ≥ 7) = 0.0219 [1]. [4] Examiner insight: quote the probabilities either side of the boundary to justify the critical value.

5. (a) Under H₀, X ~ B(50, 0.3); 2-tail, so 2.5% in each tail [1]. Lower: P(X ≤ 8) = 0.0183, P(X ≤ 9) = 0.0402, so X ≤ 8 [1]. Upper: P(X ≥ 23) = 0.0123, P(X ≥ 22) = 0.0251, so X ≥ 23 [1]. Critical region X ≤ 8 or X ≥ 23 [1]. (b) 0.0183 + 0.0123 = 0.0305 (3 s.f.) [1]. (c) 21 is not in the critical region, so do not reject H₀: there is insufficient evidence to suggest the probability of heads is not 0.3 [1]. [6] Examiner insight: P(X ≥ 22) = 0.0251 is just above 0.025, so X ≥ 22 is wrong; comparing each tail with 0.05 instead of 0.025 gives the wrong region.

6. (a) H₀: ρ = 0, H₁: ρ < 0 [1]. (b) Compare r = −0.47 with −0.4409: −0.47 < −0.4409 [1]. Reject H₀: there is sufficient evidence at the 5% level of negative correlation between age and resale value of cars in the population [1]. (c) The points lie moderately close to a straight line sloping downwards [1]. [4] Examiner insight: hypotheses must use ρ, not r; writing H₀: r = 0 loses the mark even if the comparison is correct.

7. (a) 0.069 > 0.05 [1], so do not reject H₀: there is insufficient evidence at the 5% level of correlation between rainfall and umbrella sales [1]. (b) Yes: 0.069 < 0.10, so H₀ would be rejected at 10% [1]. [3] Examiner insight: a conclusion such as “rainfall causes umbrella sales” is not credited; the test only gives evidence about correlation.

8. Let μ = mean playback time after the update. H₀: μ = 250, H₁: μ < 250 [1]. Under H₀, X̄ ~ N(250, 12²/20) [1]. z = (244.8 − 250)/(12/√20) [1] = −1.94 (3 s.f.), or P(X̄ ≤ 244.8) = 0.0263 [1]. −1.94 < −1.645 (or 0.0263 < 0.05) [1]. Reject H₀: there is sufficient evidence at the 5% level to suggest the mean playback time has decreased [1]. [6] Examiner insight: using 12 instead of 12/√20 gives z = −0.433 and the wrong conclusion; the distribution mark depends on the variance σ²/n.

9. (a) H₀: μ = 18.5, H₁: μ ≠ 18.5; under H₀, X̄ ~ N(18.5, 2.4²/36), so σ/√n = 0.4 [1]. Critical z values ±1.960 [1]. 18.5 ± 1.960 × 0.4 = 17.72 and 19.28 (2 d.p.) [1]. Critical region X̄ < 17.72 or X̄ > 19.28 [1]. (b) 19.4 > 19.28, so 19.4 is in the critical region [1]. Reject H₀: there is sufficient evidence at the 5% level to suggest the mean length of lizards on the island is different from 18.5 cm [1]. [6] Examiner insight: in a 2-tail test the z value is 1.960, not 1.645; using 1.645 gives a region that is too wide and loses the accuracy mark.

10. (a) Let p = proportion of all visitors who click after the redesign. H₀: p = 0.12, H₁: p > 0.12 [1]. Under H₀, X ~ B(40, 0.12) [1]. p-value = P(X ≥ 9) = 1 − P(X ≤ 8) [1] = 0.0443 [1]. 0.0443 < 0.05, so reject H₀: there is sufficient evidence at the 5% level to suggest the proportion who click has increased [1]. (b) P(X ≥ 11) = 0.00615 and P(X ≥ 10) = 0.0175 [1], so the critical region is X ≥ 11 [1]. (c) The probability of concluding that the proportion has increased when it is still 0.12 is at most 1% [1]. [8] Examiner insight: using 1 − P(X ≤ 9) for P(X ≥ 9) gives 0.0175 and loses the accuracy mark.

11. (a) Let μ = mean length of pins. H₀: μ = 4.2, H₁: μ > 4.2 [1]. Under H₀, X̄ ~ N(4.2, 0.5²/25) [1]. z = (4.37 − 4.2)/0.1 = 1.7 [1]. Critical value z = 1.960 at 2.5%; 1.7 < 1.960 [1]. Do not reject H₀: there is insufficient evidence at the 2.5% level to suggest the mean length has increased [1]. (b) Need (4.37 − 4.2)/(0.5/√n) > 1.960 [1]. √n > 1.960 × 0.5/0.17 = 5.765, so n > 33.2 [1]. Smallest n = 34 [1]. [8] Examiner insight: the final answer must be an integer that satisfies the inequality; rounding 33.2 down to 33 loses the accuracy mark.

Where marks are usually lost

• Hypotheses written in terms of r, x̄ or a sample proportion instead of ρ, μ or p.
• The parameter is never defined, so the hypotheses have no context.
• P(X = k) given as the p-value instead of a cumulative tail probability.
• P(X ≥ k) found as 1 − P(X ≤ k).
• In a 2-tail test, each tail compared with the full significance level.
• Critical regions stated with no probabilities either side of the boundary.
• σ used in place of σ/√n for the sample mean.
• Conclusions written as certainties, or with no context.

Next steps

• Hypothesis testing revision notes
• Hypothesis testing study guide
• Statistical sampling practice
• AQA A-level Mathematics course hub
• Printable 7357 checklist
• All free 10-minute diagnostics
• Book a free trial class

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3, 31 January 2018, for A-level exams June 2018 onwards, Section 3.16 O: Statistical hypothesis testing. Published by AQA.