| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2011 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of a Poisson distribution |
| Type | Comment on test validity or assumptions |
| Difficulty | Standard +0.3 This is a standard S3 hypothesis testing question following a routine template: state Poisson assumptions, calculate mean from grouped data, find expected frequencies using Poisson probabilities, and perform a chi-squared goodness-of-fit test. All steps are procedural with no novel insight required, making it slightly easier than average for A-level. |
| 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 = lambda5.06b Fit prescribed distribution: chi-squared test |
| No of hurricanes, \(h\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Frequency | 0 | 2 | 5 | 17 | 20 | 12 | 12 | 12 |
| \(h\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 or more |
| Expected frequency | 0.90 | 4.04 | \(r\) | 13.55 | \(s\) | 13.65 | 10.21 | 13.39 |
The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
No of hurricanes, $h$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
Frequency & 0 & 2 & 5 & 17 & 20 & 12 & 12 & 12 \\
\hline
\end{tabular}
Table 1
\begin{enumerate}[label=(\alph*)]
\item Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution.
[2]
\item Show that the mean number of hurricanes per year from Table 1 is 4.4875
[2]
\item Use the answer in part (b) to calculate the expected frequencies $r$ and $s$ given in Table 2 below to 2 decimal places.
[3]
\end{enumerate}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$h$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 or more \\
\hline
Expected frequency & 0.90 & 4.04 & $r$ & 13.55 & $s$ & 13.65 & 10.21 & 13.39 \\
\hline
\end{tabular}
Table 2
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Test, at the 5\% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly.
[6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2011 Q5 [13]}}