9231 P22 - Nov 2018 - Q10 - 12 marks
6137
The number of accidents, \(x\), that occur each day on a motorway are recorded over a period of 40 days. The results are shown in the following table.
| Number of accidents | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
|---|---|---|---|---|---|---|---|---|
| Observed frequency | 3 | 5 | 8 | 10 | 5 | 7 | 2 | 0 |
(i) Show that the mean number of accidents each day is \(2.95\) and calculate the variance for this sample. Explain why these values suggest that a Poisson distribution might fit the data.
A Poisson distribution with mean \(2.95\), as found from the data, is used to calculate the expected frequencies, correct to 2 decimal places. The results are shown in the following table.
| Number of accidents | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
|---|---|---|---|---|---|---|---|---|
| Observed frequency | 3 | 5 | 8 | 10 | 5 | 7 | 2 | 0 |
| Expected frequency | 2.09 | 6.18 | 9.11 | 8.96 | 6.61 | 3.90 | 1.92 | 1.23 |
(ii) Show how the expected frequency of \(6.61\) for \(x=4\) is obtained.
(iii) Test at the \(5\%\) significance level the goodness of fit of this Poisson distribution to the data.
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