10 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 |
- 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 |
- Show how the expected frequency of 6.61 for \(x = 4\) is obtained.
- Test at the \(5 \%\) significance level the goodness of fit of this Poisson distribution to the data.