| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | June |
| Marks | 12 |
| 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 Poisson distribution question requiring standard applications: identifying the model, calculating probabilities using tables/calculator, and applying normal approximation. Part (d) requires recognizing when to use continuity correction, which is slightly above routine but still a standard S2 technique. The question involves no novel problem-solving or conceptual challenges beyond textbook exercises. |
| Spec | 5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
| Answer | Marks | Guidance |
|---|---|---|
| No of defects in carpet area \(a\) sq m is distributed \(Po(0.05a)\). Defects occur at a constant rate, independent, singly, randomly | B1B1, Any 1 B1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X = 2) = \frac{e^{-1.5} \times 1.5^2}{2} = 0.2510\) | B1, M1A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9955 = 0.0045\) | M1M1A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| \(= 0.1867\) | B1, B1, M1M1, awrt 0.89, A1, A1 | (6) |
## Part (a)
No of defects in carpet area $a$ sq m is distributed $Po(0.05a)$. Defects occur at a constant rate, independent, singly, randomly | B1B1, Any 1 B1 | (3)
## Part (b)
$X \sim P(30 \times 0.05) = P(1.5)$
$P(X = 2) = \frac{e^{-1.5} \times 1.5^2}{2} = 0.2510$ | B1, M1A1 | (3)
Tables or calc 0.251(0)
## Part (c)
$P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9955 = 0.0045$ | M1M1A1 | (3)
Strict inequality, 1-0.9955, 0.0045
## Part (d)
$X \sim P(17.75)$
$X \sim N(17.75, 17.75)$
$P(X \geq 22) = P\left(Z > \frac{21.5 - 17.75}{\sqrt{17.75}}\right)$
$= P(Z > 0.89)$
$= 0.1867$ | B1, B1, M1M1, awrt 0.89, A1, A1 | (6)
Standardise, accept 22 or ±0.5Minor defects occur in a particular make of carpet at a mean rate of 0.05 per m$^2$.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer.
\end{enumerate}
A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires 30 m$^2$ of this carpet. Find the probability that the foyer carpet contains
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item exactly 2 defects, [3]
\item more than 5 defects. [3]
\end{enumerate}
The carpet fitter orders a total of 355 m$^2$ of the carpet for the whole hotel.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2004 Q6 [12]}}