| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| 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 straightforward calculations. Part (a) is routine mean calculation, part (b) requires basic Poisson probability calculations, and part (c) follows a standard hypothesis testing procedure. While it requires multiple steps and understanding of the chi-squared test, it's a textbook application with no novel insights needed—slightly easier than average for S3 level. |
| Spec | 2.02f Measures of average and spread5.02i 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 test5.06c Fit other distributions: discrete and continuous |
| Number of accidents | 0 | 1 | 2 | 3 | 4 | 5 |
| Frequency | 47 | 57 | 46 | 35 | 9 | 6 |
| Number of accidents | 0 | 1 | 2 | 3 | 4 | 5 or more |
| Frequency | 40.38 | 64.61 | \(r\) | 27.57 | 11.03 | \(s\) |
| Answer | Marks | Guidance |
|---|---|---|
| 3 | 35 | 27.57 |
Question 3:
3 | 35 | 27.57 | 35 | 27.57 | 2.0024 | 44.4324The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Number of accidents & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
Frequency & 47 & 57 & 46 & 35 & 9 & 6 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Show that the mean number of accidents per day for these data is 1.6 [1]
\end{enumerate}
A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution.
She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Number of accidents & 0 & 1 & 2 & 3 & 4 & 5 or more \\
\hline
Frequency & 40.38 & 64.61 & $r$ & 27.57 & 11.03 & $s$ \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $r$ and the value of $s$, giving your answers to 2 decimal places. [3]
\item Stating your hypotheses clearly, use a 10\% level of significance to test the motorway supervisor's belief. Show your working clearly. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2015 Q3 [11]}}