AQA AS Level Mathematics Paper 2, June 2025: Question 21

5 marks · Medium difficulty · Multi-step Problem

Conduct a 5% significance level binomial hypothesis test to determine if the proportion of concert attendees having an excellent experience is less than 80%, given a sample of 42 out of 60.

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Question

Question 21: The organiser of a music concert claimed that 80% of the people who attended the concert had an 'excellent' experience. To test this claim, a random sample of 60 people who attended the concert were asked about their experience, and 42 described it as 'excellent'. Investigate, at the 5% level, whether there is evidence that the proportion of people who had an 'excellent' experience at the concert is less than the organiser’s claim of 80% [5 marks].
Question text

21 The organiser of a music concert claimed that 80% of the people who attended the

concert had an ‘excellent’ experience.

To test this claim, a random sample of 60 people who attended the concert were asked

about their experience, and 42 described it as ‘excellent’.

Investigate, at the 5% level, whether there is evidence that the proportion of people who

had an ‘excellent’ experience at the concert is less than the organiser’s claim of 80%

[5 marks]

Mark scheme

Show the mark scheme Mark scheme for Question 21 listing 5 marks: B1 for stating null hypothesis H0: p = 0.8 and alternative hypothesis H1: p < 0.8; M1 for identifying the binomial distribution model X ~ B(60, 0.8) and setting up P(X <= 42); A1 for calculating P(X <= 42) = 0.0427 (awrt 0.043); A1 for comparing 0.0427 < 0.05 and concluding to reject H0; R1 for a fully correct conclusion in context stating there is sufficient evidence that the proportion is less than the organiser's claim.

Q Marking instructions AO Marks Typical solution

21 States both hypotheses 2.5 B1

correctly for a one-tailed test. X is ‘No of people who said they

Accept population proportion for had an ‘excellent’ experience’

p. Accept 80%, but not x = or

x = or μ = H0: p = 0.8

H1: p < 0.8

States model used 1.1a M1 Under H0: X ~ B(60, 0.8)

PI by AWRT 0.043, 0.022,

0.077, 0.011, 0.021 P(X ≤ 42) = 0.0427

As 0.0427 < 0.05

Obtains AWRT 0.043 1.1b A1

condone AWRT 0.022 for A1

Reject H0

Compares 0.043 to 0.05 and 3.5a A1

states rejects H0 There is sufficient evidence to

Condone accept H1 suggest that the proportion of

No ft here. Must see clear people who had an excellent

comparison (inequality or experience at the concert is less

diagram) than the organiser’s claim.

Concludes correctly in context 3.2a R1

‘sufficient evidence’ OE and

proportion required.

Can score B0 M1 A1 A1 R1

Question 21 Total 5

How to answer it

One-Tailed Binomial Hypothesis Testing

📌 What this question tests
  • Setting up null and alternative hypotheses for a one-tailed test using the population parameter p.
  • Defining and applying a binomial distribution model: X ~ B(n, p).
  • Calculating cumulative binomial probabilities using a scientific/graphic calculator: P(X ≤ x).
  • Comparing the calculated p-value against a given significance level (α = 0.05).
  • Making a formal statistical decision and communicating a non-definitive, contextual conclusion.

Question 21 Walkthrough & Solution

5 Marks • Full Hypothesis Test

📐 Step-by-Step Calculations

  1. Define the test variable & model:
    Let X be the number of people who described their experience as 'excellent'.
    Under H₀, X ~ B(60, 0.8)
  2. State hypotheses:
    H₀: p = 0.8
    H₁: p < 0.8
  3. Calculate the test statistic probability:
    Observed value: x = 42.
    Since H₁ is "<", find the tail probability in the lower direction:
    P(X ≤ 42) = 0.04273... ≈ 0.0427 (or 0.043 to 3 d.p.)
  4. Compare with significance level:
    0.0427 < 0.05 (5% significance level)
  5. Decision:
    Reject H₀ (or accept H₁).

✅ Model Answer & Marks

Let X be the number of people rating the experience as 'excellent'.
H₀: p = 0.8
H₁: p < 0.8

[B1] Correct hypotheses stating parameter p for a one-tailed test.

Under H₀, X ~ B(60, 0.8)

[M1] Stating the binomial model (or implied by correct probability).

P(X ≤ 42) = 0.0427 (AWRT 0.043)

[A1] Correct cumulative probability.

Since 0.0427 < 0.05, we reject H₀.

[A1] Explicit inequality comparison with 0.05 and decision to reject H₀.

Conclusion: There is sufficient evidence to suggest that the proportion of people who had an 'excellent' experience at the concert is less than the organiser's claim of 80%.

[R1] Complete contextual conclusion including uncertainty and the word 'proportion'.

💡 Key Knowledge

  • Parameter notation: Hypotheses must strictly be stated in terms of p (population proportion) or defined as the population proportion. Never use sample statistics like x̄ or p̂.
  • Direction of test: The phrase "less than the organiser's claim" specifies a one-tailed test in the lower tail ( p < 0.8 ).
  • Cumulative probability: Always calculate the probability of obtaining a result as extreme or more extreme than observed: for a lower-tail test, this is P(X ≤ x) , never simply P(X = x) .

🧠 Exam Technique & Tips

  • Show the comparison: Never just state "Reject H₀". Always write the explicit comparison down: 0.0427 < 0.05 . Examiners require this to award the mark.
  • Nuanced wording: Never say "This proves that less than 80% enjoyed it". Statistical tests deal with probability, not certainty. Always write: "There is sufficient evidence to suggest...".
  • Reference the context: Mention both the context (people who had an excellent experience) and the parameter being tested (proportion).

❌ Common Examiner Traps

  • Wrong parameter letter: Writing H₀: μ = 0.8 or x̄ = 0.8 loses the B1 mark instantly.
  • Point probability trap: Calculating P(X = 42) = 0.0216 instead of the cumulative probability P(X ≤ 42) .
  • Missing the comparison: Skipping the numerical comparison step ( 0.043 < 0.05 ) prevents gaining the fourth mark [A1].
  • Omission of 'proportion': Simply concluding "less people enjoyed it" instead of "the proportion of people is less" risks losing the final communication mark [R1].

Topics

Statistics · N: Statistical distributions · O: Statistical hypothesis testing

Question and mark scheme from the AQA AS Level Mathematics examination, Paper 2, June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.