A factory owner models the number of employees who use the factory canteen on any day by the distribution B(25, p). In the past the value of p was 0.8. A new menu is introduced in the canteen and the owner wants to test whether the value of p has increased. On a randomly chosen day he notes that the number of employees who use the canteen is 23.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

1. Ho: p = 0.8 H₁: p > 0.8
2. [Assuming Ho, P(X > 23) =] 25C23×0.2²×0.8²³ + 25C24×0.2¹×0.8²⁴ + 0.8²⁵ = 0.070835+0.0236118 +0.0037779
3. = 0.0982
4. 0.0982 < 0.1
5. [There is evidence to reject Ho] / There is sufficient evidence to suggest that p has increased
6. Not suitable as model does not allow for more than 25 employees to use the canteen/Not suitable as uses a sample instead of all employees/Not suitable doesn't include all employees /Not suitable as 30 is only just bigger than 25 should have used 30 OR / Suitable as owner knows that not all employees use the canteen, or similar