| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson hypothesis test |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with standard techniques: calculating P(X > 6), finding capacity requirements, and performing a hypothesis test. All parts use direct formulas or tables with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(X \sim Po(3.5)\) | B1 M1 A1 | |
| \(P(X > 6) = 1 - 0.9347 = 0.0653\) | A1 | |
| (b) \(P(X \leq 8) = 99.01\%\), so the centre must be able to cope with 8 calls, and therefore needs 16 operators | B1 M1 A1 | |
| (c) \(P(X > 10) = 0.1\%\), \(P(X > 11) = 0.03\%\), so need 11 calls | M1 A1 A1 | Total: 9 |
(a) $X \sim Po(3.5)$ | B1 M1 A1 |
$P(X > 6) = 1 - 0.9347 = 0.0653$ | A1 |
(b) $P(X \leq 8) = 99.01\%$, so the centre must be able to cope with 8 calls, and therefore needs 16 operators | B1 M1 A1 |
(c) $P(X > 10) = 0.1\%$, $P(X > 11) = 0.03\%$, so need 11 calls | M1 A1 A1 | **Total: 9**
---A centre for receiving calls for the emergency services gets an average of 3.5 emergency calls every minute. Assuming that the number of calls per minute follows a Poisson distribution,
\begin{enumerate}[label=(\alph*)]
\item find the probability that more than 6 calls arrive in any particular minute. [3 marks]
Each operator takes a mean time of 2 minutes to deal with each call, and therefore seven operators are necessary to cope with the average demand.
\item Find how many operators are required for there to be a 99\% probability that a call can be dealt with immediately. [3 marks]
It is found from experience that a major disaster creates a surge of emergency calls. Taking the null hypothesis $H_0$ that there is no disaster,
\item find the number of calls that need to be received in one minute to disprove $H_0$ at the 0.1 \% significance level. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [9]}}