| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 This is a straightforward two-part Poisson question requiring (a) direct application of the Poisson probability formula with rate adjustment, and (b) a standard one-tailed hypothesis test. Both parts follow textbook procedures with no conceptual challenges—slightly easier than average due to clear setup and routine methods. |
| Spec | 2.05e Hypothesis test for normal mean: known variance5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x! |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(X \sim \text{Po}(6)\) \(P(X = 1) = [=6e^{-6} = 0.0174 - 0.0025] = 0.01487\ldots\) awrt \(\mathbf{0.0149}\) | M1 A1 (2) |
| (b) | \(H_0: \lambda = 6\) (or 9) \(H_1: \lambda > 6\) (or 9) \(Y \sim \text{Po}(9)\) \(P(Y \geq 14) = 1 - P(Y \leq 13) = 1 - 0.9261 = 0.0739 / P(Y \geq 15) = 0.0415\), CR: \(Y \geq 15\) Do not reject \(H_0\) / Not significant / 14 is not in the critical region. There is not enough evidence to suggest that the rate of calls for reservations has increased. | B1 M1 A1 dM1 A1cso (5) Total 7 |
(a) | $X \sim \text{Po}(6)$ $P(X = 1) = [=6e^{-6} = 0.0174 - 0.0025] = 0.01487\ldots$ awrt $\mathbf{0.0149}$ | M1 A1 (2) | 1st M1 for writing or using Po(6)
(b) | $H_0: \lambda = 6$ (or 9) $H_1: \lambda > 6$ (or 9) $Y \sim \text{Po}(9)$ $P(Y \geq 14) = 1 - P(Y \leq 13) = 1 - 0.9261 = 0.0739 / P(Y \geq 15) = 0.0415$, CR: $Y \geq 15$ Do not reject $H_0$ / Not significant / 14 is not in the critical region. There is not enough evidence to suggest that the rate of calls for reservations has increased. | B1 M1 A1 dM1 A1cso (5) Total 7 | 1st B1 for both hypotheses correct with $\lambda$ or $\mu$ / 1st M1 for writing or using $1 - P(Y \leq 13)$ and Po(9) or writing or using $P(Y \geq 15)$ and Po(9) for a CR method / 1st A1 for awrt 0.0739 / CR: $Y \geq 15$ / $Y > 14$ / 2nd dM1 dependent on 1st M1 for correct statement (i.e. Do not reject $H_0$ / Not significant / 14 is not in the critical region) (may be implied by a correct contextual statement). Do not allow contradictory statements. / 2nd A1cso A correct contextual statement must include the word **calls** and the idea the rate has not increased. All previous marks must be awarded for this mark to be awarded. / SC: $1 - P(Y \leq 14) = 0.0415$ so reject $H_0$ scores M0A0M1A0
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\item Each day a restaurant opens between 11 am and 11 pm . During its opening hours, the restaurant receives calls for reservations at an average rate of 6 per hour.\\
(a) Find the probability that the restaurant receives exactly 1 call for a reservation between 6 pm and 7 pm .
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The restaurant distributes leaflets to local residents to try and increase the number of calls for reservations. After distributing the leaflets, it records the number of calls for reservations it receives over a 90 minute period.
Given that it receives 14 calls for reservations during the 90 minute period,\\
(b) test, at the $5 \%$ level of significance, whether the rate of calls for reservations has increased from 6 per hour. State your hypotheses clearly.
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\hfill \mbox{\textit{Edexcel S2 2018 Q1 [7]}}