It is known that 1 in 5000 people in Atalia have a certain condition. A random sample of 12500 people from Atalia is chosen for a medical trial. The number having the condition is denoted by X.

Mark scheme

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

1. [1] (a) Identify appropriate approximating distribution as Poisson, X ~ Po(λ), where λ = np = 12500 * (1/5000) = 2.5.
2. [1] (a) Calculate P(X ≤ 3) using Poisson formula: P(X=0) + P(X=1) + P(X=2) + P(X=3).
3. [1] (a) P(X ≤ 3) = 0.758 (3 s.f.).
4. [1] (b) For Binomial X~B(n,p), E(X) = np = 2.5 and Var(X) = np(1-p) = 2.4995. (Just 2.5 for variance is not sufficient without explanation).
5. [1] (b) Explain that E(X) is approximately equal to Var(X) (condone 'equal'), which is a characteristic of a Poisson distribution, thus confirming the approximation is appropriate.