N2007 P2 Q7

A large number of students in a college have completed a geography project. The time, \(x\) hours, taken by a student to complete the project is noted for a random sample of 150 students. The results are summarised by \(\sum x=4626\), \(\sum x^2=147691\).

Find unbiased estimates of the population mean and variance.[2]

Test, at the 5% significance level, whether the population mean time for a student to complete the project exceeds 30 hours.[4]

State, giving a reason, whether any assumptions about the population are needed in order for the test to be valid.[1]