10 Roberto owns a small hotel and offers accommodation to guests. Over a period of 100 nights, the numbers of rooms, \(x\), that are occupied each night at Roberto's hotel and the corresponding frequencies are shown in the following table.
| Number of rooms | | occupied \(( x )\) |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Number of nights | 4 | 9 | 18 | 26 | 20 | 16 | 7 | 0 |
- Show that the mean number of rooms that are occupied each night is 3.25 .
The following table shows most of the corresponding expected frequencies, correct to 2 decimal places, using a Poisson distribution with mean 3.25.
| Number of rooms | | occupied \(( x )\) |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) |
| Observed frequency | 4 | 9 | 18 | 26 | 20 | 16 | 7 | 0 |
| Expected frequency | 3.88 | 12.60 | 20.48 | 22.18 | 18.02 | 11.72 | | |
- Show how the expected value of 22.18 , for \(x = 3\), is obtained and find the expected values for \(x = 6\) and for \(x \geqslant 7\).
- Use a goodness-of-fit test at the \(5 \%\) significance level to determine whether the Poisson distribution is a suitable model for the number of rooms occupied each night at Roberto's hotel.