2025 MI P2 Q12

Supernova School is a school which specialises in sports. The mean reaction time of the students in this school is \(250\) milliseconds. To reduce the reaction times of students, the teachers implemented a reflex training programme. After its completion, \(50\) students were randomly selected from the school, and their reaction times \(x\) (in milliseconds) were recorded as follows:

\[\sum (x-200) = 2159.62 \quad \quad \sum (x-200)^2 = 112081.09\]

1. Calculate the unbiased estimates of the population mean and variance of the reaction time of students.[2]
2. Test, at \(1\%\) level of significance, whether the programme was effective. You should state your hypotheses and define any symbols you use.[5]
3. Explain why the test in part (ii) may not be appropriate if only a sample of \(10\) students was taken.[1]

In another school, Galaxy School, the reaction times of its students follow a normal distribution with mean \(\mu_0\) milliseconds and variance \(153\) milliseconds\({}^2\). The school also implemented the reflex training programme with its students.

4. A random sample of \(40\) students is taken and the mean reaction time of this sample is computed to be \(241\) milliseconds. A test carried out at \(5\%\) level of significance indicated that the reflex training programme was not successful. Find the range of values of \(\mu_0\).[4]