The number of cars arriving at a certain road junction on a weekday morning has a Poisson distribution with mean 4.6 per minute. Traffic lights are installed at the junction and a council officer wishes to test at the 2% significance level whether there are now fewer cars arriving. He notes the number of cars arriving during a randomly chosen 2-minute period.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B1] H0: Pop mean = 4.6 [or 9.2] H1: Pop mean < 4.6 [or 9.2] or λ = 4.6 or μ (Not just 'mean') or λ < 4.6
2. [B1] Use of Poisson with λ = 9.2 SOI
3. [M1] P(X ≤ 3) = e^-9.2(1 + 9.2 + 9.2^2/2 +9.2^3/3!) = 0.0184 or 0.018 [< 0.02] At least one of these attempted correct λ (with Poisson expression seen not implied)
4. [*A1] Both correct SC Use of λ = 4.6 scores B1 for P(X = 0) = 0.01[0][1] and P(X < 1) = 0.056[3]only
5. [DA1] CR is X < 3 From CWO and at least one comparison seen SC If M0 awarded allow *B1 for both 0.018 and 0.049 or better and DB1 for correct critical region from CWO and at least one comparison seen.
6. [M1] 5 is not in critical region OR P(X < 5) =0.104 > 0.02 so [not reject Ho] no evidence that number of cars arriving is now fewer For a comparison (i.e. 5 > 3) OE
7. [A1 FT] In context, not definite No contradictions e.g. not 'No. of cars arriving is not fewer' ft their critical region if used (but must be from Poisson and integers)
8. [B1 FT] No, because Ho was not rejected FT OE, FT their (c)
9. [B1] N(276, 276) SOI
10. [M1] (300.5-276)/√276 [= 1.475] Standardising with their values Allow with wrong or no continuity correction
11. [M1] 1 – Φ('1.475') = 0.0701 (3 s.f.) For area consistent with their values
12. [A1] As final answer