A new light was installed on a certain footpath. A town councillor decided to use a hypothesis test to investigate whether the number of people using the path in the evening had increased. Before the light was installed, the mean number of people using the path during any 20-minute period during the evening was 1.01. After the light was installed, the total number, n, of people using the path during 3 randomly chosen 20-minute periods during the evening was noted.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [1] (a) Hypotheses: H₀: λ = 3.03 (mean number of people for 3 periods), H₁: λ > 3.03 (mean number of people has increased).
2. [1] (a) Use Poisson distribution X ~ Po(3.03).
3. [1] (a) Calculate P(X ≥ 6) = 1 - P(X ≤ 5) using the Poisson formula with λ = 3.03.
4. [1] (a) P-value = 0.0870 (3 s.f.).
5. [1] (a) Comparison: 0.0870 > 0.05.
6. [1] (a) Conclusion: Do not reject H₀. Insufficient evidence that the mean number of people using the path has increased. Conclusion must be in context and not definite.
7. [1] (b) Critical region identified as X ≥ 7 (from P(X ≥ c | λ=3.03) ≤ 0.05). This can be calculated as P(X ≥ 6) - P(X = 6).
8. [1] (b) Probability of a Type I error = P(X ≥ 7 | λ=3.03) = 0.0350 or 0.0351 (AWRT 0.035).
9. [1] (c) A Type I error is when H₀ is rejected when it is actually true. In this context, it means concluding that the mean number of people using the path has increased, when in fact it has not (i.e., it is still 3.03 or less).
10. [1] (d) A value for the true mean (under H₁).
11. [1] (d) This true mean should be specified for the 3 randomly chosen 20-minute periods in the evening (i.e., the mean number of people using the path per 20 mins).