2025 TJC P2 Q9

At a school canteen, the average queuing time (in minutes) for a student to be served at a stall is known to be \(4.5\) minutes. The school operation manager claims that with a recent change in students’ timetable, there is an improvement in the average queuing time.

A random sample of \(10\) students’ queuing time is observed after the change in students’ timetable and the data is shown below.

\[4.2, \quad 5.2, \quad 3.1, \quad 3.5, \quad 4.6, \quad 4.8, \quad 3.8, \quad 3.7, \quad 4.5, \quad 5.0\]

Given that the population variance of queuing time is known to be \(0.48\) minutes\(^2\), a one-tailed test is carried out to determine whether the manager’s claim is justified.

1. State an assumption needed to carry out the hypothesis test.[1]
2. Test whether the manager’s claim is justified at \(5\)% level of significance.[5]

A student leader Jason suspected that with the change in students’ timetable, the average queuing time for a student to be served at a stall is in fact more than \(\lambda\) minutes and also that the population variance of queuing time is no longer \(0.48\) minutes\(^2\). To investigate this, Jason recorded the queuing time of the first \(50\) students who visited the stall from \(12\) pm to \(1\) pm.

3. Give a reason why the sample collected is not a random sample.[1]

Jason decided to collect another set of data to do his investigation. He recorded the queuing time, \(x\), in minutes, of a random sample of \(50\) students. The summarized data is as shown below.

\[\bar{x} = 4.6 \quad \text{and} \quad \sum x^2 = 1068\]

4. Find an unbiased estimate of the population variance.[1]

Jason carried out a hypothesis test at \(2\)% level of significance with an alternative hypothesis of \(\mu > \lambda\) where \(\mu\) is the population mean queuing time.

5. Explain why the assumption in (a) is no longer necessary.[1]
6. By finding the critical region in terms of \(z\)-values, find the range of values of \(\lambda\) such that the Jason’s suspicion is justified.[3]