2019 DHS P2 Q9

The time taken, \(T\) (in minutes), for a \(17\)-year-old student to complete a 5km run is random vairbale with mean \(30\). After a new training programme is introduced for these students, a random sample of \(n\) students is taken. The mean time and standard deviation for the sample are found to be \(28.9\) minutes and \(4.0\) minutes respectively.

1. Find the unbiased estimate of the population variance in terms of \(n\).[1]
2. Using \(n=40\),
   1. carry out a test at the 10% significance level to determine if the mean time taken has changed. State appropriate hypotheses for the test and define any symbols you use.[4]
   2. State what it means by the \(p\)-value in this context.[1]
   3. Give a reason why no assumptions about the population are needed in order for the test to be valid.[1]
3. The trainer claims instead that the new training programme is able to improve the mean of \(T\), \(30\) minutes, by at least \(5\)%. The school wants to test his claim.
   1. Write down the null and alternative hypothesis.[1]
   2. Using the existing sample, the school carried out a test at \(1\)% significance level and found that there was sufficient evidence to reject the trainer’s claim. Find the set of values that \(n\) can take, stating any necessary assumption(s) needed to carry out the test.[4]