2019 EJC BT2 P2 Q8 [Modified]

A private-hire vehicle firm, Euber, employs drivers to operate a fleet of cars. The mileage covered by an Euber driver on a weekday is denoted by \(M\) km. A random sample of \(60\) Euber drivers is taken and their mileage can be summarised by

\[\sum(m-250) = 540 \text{ and}\hspace{0.5em} \sum(m-259)^2 = 8176.\]

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

Assume that the population mean and variance follow the answers calculated in part (i).

Random samples of \(n\) Euber drivers are taken, where \(n\) is large.

2. Over a long period of time, it is found that \(25\%\) of the samples have a mean mileage of at least \(260\) km. Find the value of \(n\).[3]
3. Explain whether it is necessary to assume that \(M\) follows a normal distribution in order for your calculations in part (ii) to hold.[1]
4. It is now given that \(M\) is normally distributed with population variance \(5^2\). Estimate the sample size needed so that Euber can be \(95\%\) certain that the sample mean mileage will differ from the true mean by less than \(500\)m.[2]