| Exam Board | OCR |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Poisson to the Normal distribution |
| Type | Combined Poisson approximation and exact calculation |
| Difficulty | Standard +0.3 This is a straightforward S2 question combining routine Poisson probability calculations (using tables or calculator) with a standard normal approximation to Poisson for large λ. Part (iii) tests conceptual understanding of Poisson conditions but requires only textbook knowledge. All techniques are standard with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(1 - 0.8153 = 0.1847\) | M1, A1 2 | Po(3) tables, "1 −" used, e.g. 0.3528 or 0.8153; Answer 0.1847 or 0.185 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.8153 - 0.6472 = 0.168\) | M1, A1 2 | Subtract 2 tabular values, or formula \([e^{-3} \cdot 3^4/4!]\); Answer a.r.t. 0.168 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(N(150, 150)\) | B1 | Normal, mean \(3\times50\) stated or implied |
| \(1 - \Phi\!\left(\dfrac{165.5-150}{\sqrt{150}}\right)\) | B1, M1, A1 | Variance or SD \(= 3\times50\); Standardise 165 with \(\lambda\), \(\sqrt{\lambda}\) or \(\lambda\); \(\sqrt{\lambda}\) and 165.5 |
| \(= 1 - \Phi(1.266) = \mathbf{0.103}\) | A1 5 | Answer in range [0.102, 0.103] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (a) The sale of one house does not affect the sale of any others | B1 | Relevant answer showing evidence of correct understanding [but *not* just examples] |
| (b) The average number of houses sold in a given time interval is constant | B1 2 | Different reason, in context. [Allow "constant rate" or "uniform" but not "number constant", "random", "singly", "events".] |
# Question 6:
## Part (i)(a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $1 - 0.8153 = 0.1847$ | M1, A1 **2** | Po(3) tables, "1 −" used, e.g. 0.3528 or 0.8153; Answer 0.1847 or 0.185 |
## Part (i)(b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.8153 - 0.6472 = 0.168$ | M1, A1 **2** | Subtract 2 tabular values, or formula $[e^{-3} \cdot 3^4/4!]$; Answer a.r.t. 0.168 |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $N(150, 150)$ | B1 | Normal, mean $3\times50$ stated or implied |
| $1 - \Phi\!\left(\dfrac{165.5-150}{\sqrt{150}}\right)$ | B1, M1, A1 | Variance or SD $= 3\times50$; Standardise 165 with $\lambda$, $\sqrt{\lambda}$ or $\lambda$; $\sqrt{\lambda}$ and 165.5 |
| $= 1 - \Phi(1.266) = \mathbf{0.103}$ | A1 **5** | Answer in range [0.102, 0.103] |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| (a) The sale of one house does not affect the sale of any others | B1 | Relevant answer showing evidence of correct understanding [but *not* just examples] |
| (b) The average number of houses sold in a given time interval is constant | B1 **2** | Different reason, in context. [Allow "constant rate" or "uniform" but not "number constant", "random", "singly", "events".] |
---6 The number of house sales per week handled by an estate agent is modelled by the distribution $\operatorname { Po } ( 3 )$.\\
(i) Find the probability that, in one randomly chosen week, the number of sales handled is
\begin{enumerate}[label=(\alph*)]
\item greater than 4 ,
\item exactly 4 .\\
(ii) Use a suitable approximation to the Poisson distribution to find the probability that, in a year consisting of 50 working weeks, the estate agent handles more than 165 house sales.\\
(iii) One of the conditions needed for the use of a Poisson model to be valid is that house sales are independent of one another.\\
(a) Explain, in non-technical language, what you understand by this condition.\\
(b) State another condition that is needed.
\end{enumerate}
\hfill \mbox{\textit{OCR S2 2008 Q6 [11]}}