2025 CJC P2 Q11

EastHam is a football club in a competitive league. It is found that the average time taken by EastHam to score their first goal in a match is \(50\) minutes. The coach trialled a new play tactic and the time taken, \(x\) minutes, to score the first goal is recorded for a random sample of \(35\) matches in the season. The total time taken is found to be \(1650\) minutes and the variance of the time is \(250\) minutes\(^2\).

The coach wants to test if the average time taken to score the first goal has decreased with the implementation of the new play tactic.

1. Explain why the coach is able to carry out a hypothesis test without any assumption about the distribution of the time taken to score the first goal.[1]
2. Find unbiased estimates of the population mean and variance. Give your answers correct to \(3\) decimal places.[2]
3. Carry out the test at \(10\%\) level of significance. You should state your hypotheses and define any symbols you use.[4]

A fan of another football club in the same league, NewPalace, claims that the average time taken by NewPalace to score their first goal in a match is \(40\) minutes. The time taken by NewPalace to score the first goal in a match can be assumed to be normally distributed with variance \(280\) minutes\(^2\).

4. A random sample of \(50\) matches is taken and a hypothesis test is carried out on whether the fan's claim is valid. Find the range of values of the mean time of this sample for which the fan's claim would be rejected at \(10\%\) level of significance.[4]