7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.
One hour is subtracted from each of his actual times, and the resulting differences, \(x\) minutes, for a random sample of 100 boiler services are summarised in the table.
| Difference | Frequency |
| \(- 6 \leqslant x < - 4\) | 4 |
| \(- 4 \leqslant x < - 2\) | 9 |
| \(- 2 \leqslant x < 0\) | 13 |
| \(0 \leqslant x < 2\) | 27 |
| \(2 \leqslant x < 4\) | 21 |
| \(4 \leqslant x < 6\) | 15 |
| \(6 \leqslant x < 8\) | 7 |
| \(8 \leqslant x \leqslant 10\) | 4 |
| Total | 100 |
- Calculate estimates of the mean and the standard deviation of these differences.
(4 marks) - Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
- Construct an approximate \(98 \%\) confidence interval for the mean time taken by Vernon to carry out a boiler service.
- Give a reason why this confidence interval is approximate rather than exact.
- Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour.
Comment, with a justification, on each of these claims.