(a) The proportion of people having a particular medical condition is 1 in 100000. A random sample of 2500 people is obtained. The number of people in the sample having the condition is denoted by X. (b) The percentage of people having a different medical condition is thought to be 30%. A researcher suspects that the true percentage is less than 30%. In a medical trial a random sample of 28 people was selected and 4 people were found to have this condition.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [B1] Po(0.025) (For Poisson and correct parameter.)
2. [B1] n = 2500 > 50, np = 0.025 < 5 (Must show 2500 and 0.025. Accept p = 1/100000 < 0.1 in place of np = 0.025 < 5.)
3. [M1] 1 - e^-0.025 (Allow any λ. FT their (a)(i) if normal; must have continuity correction.)
4. [A1] 0.0247 (3sf) (Must be from Poisson. Unsupported correct answer scores B1 instead of M1 A1.)
5. [B1] H₀: p = 0.3, H₁: p < 0.3
6. [M1] 0.7^28 + 28C1 x 0.7^27 x 0.3 + 28C2 x 0.7^26 x 0.3^2 + 28C3 x 0.7^25 x 0.3^3 + 28C4 x 0.7^24 x 0.3^4 (Use of B(28, 0.3). Addition of terms must be intended. Allow one term wrong or omitted or extra.)
7. [A1] 0.0474 (Unsupported correct answer scores B1 instead of M1 A1.)
8. [M1] 0.0474 > 0.02 [Not reject Ho] (Valid comparison.)
9. [A1ft] No evidence that suspicion is true. (Not definite e.g. not ‘Suspicion is not true’, in context, no contradictions. SC use of N(8.4, 5.88) leading to 0.054 > 0.2 OE can score B1 only for comparison and correct conclusion. Correct hypotheses with p will also score B1.)