| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Single period normal approximation - large lambda direct |
| Difficulty | Standard +0.3 This is a straightforward application of standard Poisson distribution theory and normal approximation. Part (a) requires stating the Poisson model with standard justifications (independent events, constant rate). Part (b) is direct substitution into the Poisson formula. Part (c) applies the routine normal approximation to Poisson with continuity correction. All steps are textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\therefore X \sim P_0(7)\) | must be in part a context needed once | B1; B1 |
| Sales occur independently/randomly, singly, at a constant rate | B1; B1 | |
| (b) \(P(X = 5) = P(X \leq 5) - P(X \leq 4)\) or \(\frac{7^5 e^{-7}}{5!}\) | M1 | |
| \(= 0.3007 - 0.1730 = 0.1277\) | awrt 0.128 | A1 |
| (c) \(P(X > 181) \approx P(Y \geq 181.5)\) where \(Y \sim N(168, 168)\) | \(N(168, 168)\) | B1 |
| \(= P\left(z \geq \frac{181.5 - 168}{\sqrt{168}}\right)\) | \(\pm 0.5\) stand with \(\mu\) and \(\sigma\) | M1; M1 |
| Give A1 for 1.04 or correct expression | A1 | |
| \(= P(z \geq 1.04)\) | A1 | |
| \(= 1 - 0.8508\) | attempt correct area \(1-p\) where \(p > 0.5\) | M1 |
| \(= 0.1492\) | awrt 0.149 | A1 |
**(a)** $\therefore X \sim P_0(7)$ | must be in part a context needed once | B1; B1
Sales occur independently/randomly, singly, at a constant rate | B1; B1 |
**(b)** $P(X = 5) = P(X \leq 5) - P(X \leq 4)$ or $\frac{7^5 e^{-7}}{5!}$ | M1 |
$= 0.3007 - 0.1730 = 0.1277$ | awrt 0.128 | A1 |
**(c)** $P(X > 181) \approx P(Y \geq 181.5)$ where $Y \sim N(168, 168)$ | $N(168, 168)$ | B1
$= P\left(z \geq \frac{181.5 - 168}{\sqrt{168}}\right)$ | $\pm 0.5$ stand with $\mu$ and $\sigma$ | M1; M1
Give A1 for 1.04 or correct expression | A1 |
$= P(z \geq 1.04)$ | A1 |
$= 1 - 0.8508$ | attempt correct area $1-p$ where $p > 0.5$ | M1 |
$= 0.1492$ | awrt 0.149 | A1 |An estate agent sells properties at a mean rate of 7 per week.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable model to represent the number of properties sold in a randomly chosen week. Give two reasons to support your model. [3]
\item Find the probability that in any randomly chosen week the estate agent sells exactly 5 properties. [2]
\item Using a suitable approximation find the probability that during a 24 week period the estate agent sells more than 181 properties. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2006 Q3 [11]}}