A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table.
| Number of letters | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
| Number of days | 57 | 60 | 53 | 25 | 4 | 1 | 0 |
It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion.
The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.
| Number of letters | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
| Expected number of days | 53.964 | 70.693 | \(p\) | \(q\) | 6.622 | 1.735 | 0.463 |
- Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
- Carry out a goodness of fit test at the \(10 \%\) significance level.