A school decided to introduce a new P$.$E$.$ programme for its new students

to try to improve the fitness of the students. In order to see whether the

programme was effective, several tests were done. For one of these, the

students were timed on a run of $1$ kilometre in their first week in the school

and again ten weeks later. A random sample of $100$ of the students did both

runs. The differences of their mean times, subtracting the time of the second

run from that of the first, were calculated. The mean and standard deviation

were found to be $0\mathrm{.}75$ minutes and $1\mathrm{.}62$ minutes respectively.

Calculate a $90\%$ confidence interval for the population mean difference. You

may assume that the differences are distributed normally. What assumption

have you made in finding this confidence interval?

The organiser of the programme considers that it should lead to an

improvement of at least half a minute in the average times. Explain whether

or not this aim has been achieved