Date | May 2022 | Marks available | 4 | Reference code | 22M.1.AHL.TZ1.15 |
Level | Additional Higher Level | Paper | Paper 1 | Time zone | Time zone 1 |
Command term | Find | Question number | 15 | Adapted from | N/A |
Question
The number of cars arriving at a junction in a particular town in any given minute between 9:00 am and 10:00 am is historically known to follow a Poisson distribution with a mean of 5.4 cars per minute.
A new road is built near the town. It is claimed that the new road has decreased the number of cars arriving at the junction.
To test the claim, the number of cars, X, arriving at the junction between 9:00 am and 10:00 am on a particular day will be recorded. The test will have the following hypotheses:
H0 : the mean number of cars arriving at the junction has not changed,
H1 : the mean number of cars arriving at the junction has decreased.
The alternative hypothesis will be accepted if X≤300.
Assuming the null hypothesis to be true, state the distribution of X.
Find the probability of a Type I error.
Find the probability of a Type II error, if the number of cars now follows a Poisson distribution with a mean of 4.5 cars per minute.
Markscheme
X~Po(324) A1
Note: Both distribution and mean must be seen for A1 to be awarded.
[1 mark]
P(X≤300) (M1)
=0.0946831…≈0.0947 A1
[2 marks]
(mean number of cars =) 4.5×60=270 (A1)
P(X>300 λ=270) (M1)
Note: Award M1 for using λ=270 to evaluate a probability.
P(X≥301) OR 1-P(X≤300) (M1)
=0.0334207…≈0.0334 A1
[4 marks]
Examiners report
Part (a) should have been routine as all the information needed to answer it was there in the question but here again a reliance of the use of a calculator’s probability distribution functions has meant that simply stating a distribution is too frequently neglected. Many candidates failed to progress beyond part (a). In parts (b) and (c), a lack of knowledge of Type I and Type II errors prevented candidates from tackling what was otherwise a relatively straightforward question to answer. Some had difficulty with the mechanics of using their own GDC model where P(X>300) must be interpreted as either P(X≥301) or 1-P(X≤300) to be able to perform the calculation.