2024 CJC H1 Q11

A school claims that students spend an average of \(2\) hours watching TickTak per day. The times, \(x\) hours, spent watching TickTak per day by each of \(100\) randomly chosen students are summarized by

\[\sum x = 255.2 \quad \sum x^2 = 1525.\]

1. Calculate unbiased estimates of the population mean and variance for the time spent by students watching TickTak per day.[2]
2. Test, at the \(5\)% significance level, whether the school's claim is justified.[4]
3. State, with a reason, whether it is necessary to assume that the time spent watching TickTak per day are distributed normally for this test to be valid.[1]

Another random sample of \(50\) students is taken with mean time watching TickTak per day as \(k\) hours. A test at the \(2\)% significance level gives sufficient evidence that mean time is overstated by the school.

4. By using the unbiased estimate of the population variance found in (a), find the set of possible values of \(k\).[3]