3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, \(X\) minutes, is the time between an alarm being raised and the fire engine leaving the station.
The value of \(X\) was recorded on a random sample of 50 occasions. The results are summarised below, where \(\bar { x }\) denotes the sample mean.
$$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$
- Find values for the mean and standard deviation of this sample of 50 call-out times.
- Hence construct a \(99 \%\) confidence interval for the mean call-out time.
- The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than \(6 \frac { 1 } { 2 }\) minutes.
Comment on each of these claims.