Home / IB Mathematics AHL 4.18 Test for proportion AI HL Paper 1- Exam Style Questions

IB Mathematics AHL 4.18 Test for proportion AI HL Paper 1- Exam Style Questions- New Syllabus

IB DP Maths AI HL Paper 1 - All Topics

Question

Maan is investigating the frequency of car arrivals at a specific junction and assumes they follow a Poisson distribution. He performs a hypothesis test with \(H_0\): \(\mu = 12.4\) cars per minute against \(H_1\): \(\mu > 12.4\) cars per minute. During a \(10\)-minute observation period, he decides to reject the null hypothesis if more than \(130\) cars are recorded.
(a) (i) Define what is meant by a Type I error in the context of this study.
(ii) Calculate the probability of committing a Type I error for this specific test.
(b) Provide one practical reason why the Poisson distribution might be an inappropriate model for the arrival of cars at this junction.

Most-appropriate topic codes:

• AHL 4.18: Type I errors including calculations of their probabilities — part (a)
• AHL 4.17: Validity of the Poisson distribution as a model — part (b)
▶️ Answer/Explanation
Detailed solution

(a)
(i) A Type I error occurs if Maan rejects the null hypothesis (concluding the mean arrival rate is greater than \(12.4\)) when the null hypothesis is actually true (the rate is exactly \(12.4\)).
(ii) Under the assumption that \(H_0\) is true, the mean rate for a \(10\)-minute period is \(\lambda = 12.4 \times 10 = 124\) cars.
Let \(X\) be the number of cars in \(10\) minutes, where \(X \sim Po(124)\).
The probability of a Type I error is \(P(X > 130) = 1 – P(X \le 130)\).
Using a GDC: \(1 – 0.7235\dots = 0.2764\dots\)
Probability \(\approx 0.276\).

(b)
The Poisson model assumes independent arrivals and a constant average rate. In reality, arrivals may not be independent due to traffic signals (causing cars to arrive in clusters) or the rate may fluctuate significantly depending on the time of day (e.g., peak vs. off-peak hours).

More DP Math AI HL Paper 1 Questions..
Scroll to Top