A particular bowling club has a large number of members. Their ages may be modelled by a normal random variable, \(X\), with standard deviation 7.5 years.
On 30 June 2010, Ted, the club secretary, concerned about the ageing membership, selected a random sample of 16 members and calculated their mean age to be 65.0 years.
Carry out a hypothesis test, at the \(5 \%\) level of significance, to determine whether the mean age of the club's members has changed from its value of 61.4 years on 30 June 2000.
Comment on the likely number of members who were under the age of 25 on 30 June 2010, giving a numerical reason for your answer.
During 2011, in an attempt to encourage greater participation in the sport, the club ran a recruitment drive.
After the recruitment drive, the ages of members of the bowling club may be modelled by a normal random variable, \(Y\) years, with mean \(\mu\) and standard deviation \(\sigma\). The ages, \(y\) years, of a random sample of 12 such members are summarised below.
$$\sum y = 702 \quad \text { and } \quad \sum ( y - \bar { y } ) ^ { 2 } = 88.25$$
Construct a \(90 \%\) confidence interval for \(\mu\), giving the limits to one decimal place.
Use your confidence interval to state, with a reason, whether the recruitment drive lowered the average age of the club's members.