2025 EJC P2 Q9

The students in Eunoia Junior College (EJC) have been complaining of long queue times at the canteen during peak hours. It was claimed that the average queue time per student is \(20\) minutes. It is known that the standard deviation of the queue times is \(3.8\) minutes.

The admin manager of EJC wishes to test if the average queue time is in fact \(20\) minutes. She examines a random sample of \(15\) students to determine the average queue time.

1. State what it means for a sample to be random in this context.[1]
2. Given that the admin manager concludes that there is no reason to reject the null hypothesis at the \(5\)% level of significance, find the range of possible values of the sample mean, and state an assumption needed for your calculations.[4]

To reduce the queue time, a lunch pre-order system, EuOrder, was introduced. The admin manager is tasked to find out if there is any improvement to the average queue time per student. She obtained a random sample of \(50\) students and recorded their queue times, \(t\) minutes, on a particular day. The results are summarised as follows:

\[\sum (t-20) = -35 \quad \sum (t-20)^2 = 489.\]

3. Find unbiased estimates of the population mean and variance.[2]
4. Test, at the \(10\)% level of significance, the claim that EuOrder has reduced the average queue time at the canteen.[4]