2003 AJC P2 Q9 [Modified]

The times taken by two runners \(A\) and \(B\) in a \(400\)m race are independent and normally distributed. The times (in seconds) taken by \(A\) has mean \(46\) and standard deviation \(0.5\) and the times taken by \(B\) has mean \(46.2\) and standard deviation \(0.8\).

1. \(S\) denotes the total time taken by \(A\) on \(50\) randomly chosen races and \(T\) denotes the total time taken by \(B\) on \(90\) randomly chosen races. Find the mean and variance of \(S-T\).[4]
2. Let \(\overline{X}\) denotes the average time for \(A\) to run the \(400\)m track on \(10\) different occasions and \(\overline{Y}\) denotes the average time for \(B\) to run the same track on \(16\) different occasions. Find the value of \(t\) such that \(\mathrm{P}\left( \left| \overline{X}-\overline{Y}+0.2 \right|>t \right)=0.1\).[6]