| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Frequency distribution and Poisson fit |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard Poisson distribution concepts. Part (a) requires routine calculation of mean and variance from a frequency table. Part (b) asks for recognition that equal mean and variance suggests a Poisson model. Part (c) involves basic Poisson probability calculations and comparison. While multi-part with 11 marks total, all steps follow textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation5.02i Poisson distribution: random events model |
| Number of currants, \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Number of biscuits | 4 | 9 | 11 | 8 | 4 | 3 | 1 |
| Answer | Marks |
|---|---|
| (a) Mean \(= \frac{92}{40} = 2.3\) and Variance \(= \frac{300}{40} - 2.3^2 = 2.21\) | M1 A1 M1 A1 |
| (b) Poisson, because mean \(\approx\) variance, and data positively skewed | B1 B1 |
| (c) If mean \(= 2.3\), \(P(X \geq 2) = 1 - e^{-2.3} - 2.3e^{-2.3} = 0.669\) | M1 A1 |
| If mean \(= 1.9\), \(P(X \geq 2) = 1 - e^{-1.9} - 1.9e^{-1.9} = 0.566\) | M1 A1 |
| More likely to get at least 2 currants with the first machine | A1 |
| 11 |
(a) Mean $= \frac{92}{40} = 2.3$ and Variance $= \frac{300}{40} - 2.3^2 = 2.21$ | M1 A1 M1 A1 |
(b) Poisson, because mean $\approx$ variance, and data positively skewed | B1 B1 |
(c) If mean $= 2.3$, $P(X \geq 2) = 1 - e^{-2.3} - 2.3e^{-2.3} = 0.669$ | M1 A1 |
If mean $= 1.9$, $P(X \geq 2) = 1 - e^{-1.9} - 1.9e^{-1.9} = 0.566$ | M1 A1 |
More likely to get at least 2 currants with the first machine | A1 |
| | 11 |In a packet of 40 biscuits, the number of currants in each biscuit is as follows
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Number of currants, $x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Number of biscuits & 4 & 9 & 11 & 8 & 4 & 3 & 1 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find the mean and variance of the random variable $X$ representing the number of currants per biscuit. [4 marks]
\item State an appropriate model for the distribution of $X$, giving two reasons for your answer. [2 marks]
\end{enumerate}
Another machine produces biscuits with a mean of 1.9 currants per biscuit.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q5 [11]}}