2013 RI P2 Q8

The manufacturer of a particular type of battery claims that the lifetime is distributed with mean \(150\) hours and standard deviation \(16.877\) hours. A random sample of \(n\) batteries, where \(n\) is large, is taken and the lifetime, \(t\) hours, of each battery is measured. It is found that \(\sum{t}=10\mathrm{ }317\).A test is carried out to determine whether the manufacturer has overstated the mean lifetime of the batteries, and the \(p\)-value is found to be \(0.0975\).

1. Write down appropriate hypotheses for the test, defining any symbols you use. State, in context, the conclusion of the test if the level of significance is \(8\%\).[3]
2. Show that \(n\) satisfies the equation \(\frac{10317}{n}+\frac{21.871}{\sqrt{n}}-150=0\), and solve it numerically.[5]
3. State, with a reason, whether it is necessary to assume that the lifetime of the batteries is normally distributed in order to carry out the calculation in (ii).[1]