| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Mean-variance comparison for Poisson validation |
| Difficulty | Standard +0.3 This is a standard S2 Poisson distribution question requiring routine calculations: verifying mean≈variance for Poisson validity, using tables/calculator for probability calculations, and applying the sum of independent Poisson distributions. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda5.02n Sum of Poisson variables: is Poisson |
I'm ready to help clean up the mark scheme content. However, I don't see the actual mark scheme text for Question 5 in your message. You've provided "Question 5: 5" but no marking points or content to convert.
Please paste the full mark scheme content that you'd like me to clean up, and I'll convert the unicode symbols to LaTeX notation, preserve all marking annotations (M1, A1, B1, DM1, etc.), and format it clearly with one marking point per line.5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, $X$, within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data.
$$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$
Peter believes that the distribution of $X$ can be modelled by a Poisson distribution with $\lambda = 3.2$.
\begin{enumerate}[label=(\alph*)]
\item Use the summarised data to support Peter's belief.
\item Using Peter's model, calculate the probability that:
\begin{enumerate}[label=(\roman*)]
\item a single placing of the quadrat contains more than 5 quartz pebbles;
\item a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
\item when the quadrat is placed twice, at least one placing contains no quartz pebbles.
\end{enumerate}\item Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat.
Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.\\[0pt]
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2014 Q5 [14]}}