2022 NJC P2 Q9

Osoi is frequently late for school. His mean travelling time to school is \(24.5\) minutes. The Year Head asks Osoi to take an alternative route to school and to record his travelling time, \(t\) minutes, from home to school every morning on the alternative route. The results for \(72\) randomly chosen mornings when Osoi took the alternative route are summarised as follows:

\({\sum(t - 20) = 215\text{ ;}} \qquad \sum(t - 20)^2 = 3234\)

1. Calculate unbiased estimates of the population mean and variance of the travelling time for the alternative route.[2]
2. Using a \(5\%\) level of significance, test whether Osoi’s mean travelling time from home to school has shortened after taking the alternative route to school. You should state your hypotheses and define any symbols that you use.[5]

Suppose the Year Head wants to investigate if the alternative route to school has made a difference to Osoi’s mean travelling time in the morning instead of whether it has shortened his mean travelling time in the morning, at the same level of significance.

3. Find the critical region for this test.[3]