2025 YIJC P2 Q10

A company advertises that its rechargeable batteries can run non-stop for an average of \(505\) minutes after a full charge. \(100\) batteries are randomly selected, and their running time after a full charge, \(x\) minutes, are measured. The data obtained is summarised by\[\sum (x - 500) = 320, \quad \sum (x - 500)^2 = 8416.\]

1. Suggest a reason why, in this context, the given data is summarised in terms of \((x - 500)\) rather than \(x\).[1]
2. Calculate unbiased estimates of the population mean and variance of the running time of the batteries.[2]

3. A hypothesis test is carried out at the \(5\%\) level of significance. Determine whether the company is overestimating the average running time of the batteries.

   [4]
4. State, giving a reason, whether it is necessary to assume a normal distribution for the test to be valid.[1]

5. After a change is made in the manufacturing process, another test at the \(5\%\) significance level is carried out using a new random sample of size \(50\). The sample standard deviation is found to be \(8.5\) minutes. Find the range of values of the sample mean in order to conclude that the average running time of the batteries after a full charge is more than \(505\) minutes.

   [4]