2023 DHS P2 Q10

The average waiting time (in minutes) for a customer to be served at a fast-food restaurant is \(k\) minutes. The manager claims that with subsequent increased staff strength, the restaurant can now better serve the customers and the average waiting time has improved. The population variance of waiting time is known to be \(0.22\) minutes\(^{2}\).

The observed waiting times (in minutes) of a random sample of \(9\) customers after the increased staff strength are:

\({3.6, \qquad 2.5, \qquad 2.4, \qquad 3.5, \qquad 2.8, \qquad 2.9, \qquad 3.1, \qquad 3.7, \qquad 3.0}\)

1. Find the unbiased estimates of the population mean and variance of the waiting time.[2]
2. Using the value \(k=3.3\).
   1. Carry out a one-tailed test at \(5\%\) level of significance to determine whether the manager’s claim is valid, stating clearly the \(p\)-value and any assumption(s) you have made.[5]
   2. Explain the meaning of \(p\)-value in the context of the question.[1]
3. Using the same assumption in part (b)(i), a one-tailed test was carried out at \(10\%\) and it was found that the manager’s claim was justified. Find the range of possible values of \(k\).[4]