| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Poisson |
| Difficulty | Standard +0.3 This is a standard chi-squared goodness of fit test with a Poisson distribution. Part (a) requires routine calculation of Poisson probabilities, part (b) is a textbook application of the chi-squared test with cell pooling, and part (c) tests understanding of degrees of freedom. While it requires multiple steps and careful execution, it follows a completely standard template with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.06b Fit prescribed distribution: chi-squared test |
| Number of daisies | Observed frequency | Expected frequency |
| 0 | 8 | 13.53 |
| 1 | 32 | 27.07 |
| 2 | 27 | \(r\) |
| 3 | 18 | \(s\) |
| 4 | 10 | 9.02 |
| 5 | 3 | 3.61 |
| 6 | 1 | 1.20 |
| 7 | 0 | 0.34 |
| \(\geq 8\) | 1 | \(t\) |
6. An area of grass was sampled by placing a $1 \mathrm {~m} \times 1 \mathrm {~m}$ square randomly in 100 places. The numbers of daisies in each of the squares were counted. It was decided that the resulting data could be modelled by a Poisson distribution with mean 2. The expected frequencies were calculated using the model.
The following table shows the observed and expected frequencies.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Number of daisies & Observed frequency & Expected frequency \\
\hline
0 & 8 & 13.53 \\
\hline
1 & 32 & 27.07 \\
\hline
2 & 27 & $r$ \\
\hline
3 & 18 & $s$ \\
\hline
4 & 10 & 9.02 \\
\hline
5 & 3 & 3.61 \\
\hline
6 & 1 & 1.20 \\
\hline
7 & 0 & 0.34 \\
\hline
$\geq 8$ & 1 & $t$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find values for $r , s$ and $t$.
\item Using a $5 \%$ significance level, test whether or not this Poisson model is suitable. State your hypotheses clearly.
An alternative test might have been to estimate the population mean by using the data given.
\item Explain how this would have affected the test.\\
(2)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2006 Q6 [13]}}