2022 ASRJC P2 Q8

In a public swimming centre, the time spent by a randomly chosen user in using its facilities is \(T\) minutes, is known to be normally distributed. The centre manager claims that its users spend an average of \(50\) minutes to use its facilities. To check this claim, time spent by a random sample of \(60\) users were recorded. The data recorded has an average of \(47\) minutes and a standard deviation of \(16.4\) minutes.

1. Find an unbiased estimate of the population variance, giving your answer correct to \(2\) decimal places.[1]
2. Test, at the \(5\%\) significance level, whether the centre manager overstated the average time spent.[4]
3. Another sample of size \(n\left( n>30 \right)\) that was collected independently is now used to test, at the \(5\%\) significance level, whether the centre manager’s claim is valid. For this sample, the mean time taken is \(46\) minutes. If the result of the test using this information and the unbiased estimate of the population variance in part (i) is that the null hypothesis is rejected, find the least possible value that \(n\) can take.[4]