In this question you should state the parameters of any distribution you use.
The times, in minutes, taken for male runners to complete a marathon follow the distribution \(\mathrm{N}(196, 24^2)\).
Calculate the expected number of male runners who take more than \(180\) minutes to complete a marathon in a randomly chosen batch of \(80\) male runners.[2]
It is given that at most \(10\%\) of the fastest male runners will be eligible to join the competition. Find the qualifying time, to the nearest minute, to join the competition.[1]
The times, in minutes, taken for female runners to complete a marathon follow the distribution \(\mathrm{N}(210, 30^2)\).
Find the probability that the total time taken by a randomly selected male runner and \(3\) randomly selected female runners is between \(700\) and \(800\) minutes.[3]
To help the group of marathon runners improve their timings for the actual competition, a sponsor provides all runners with a set of running apparel to help them reduce air drag. This reduces the timing of each male runner by \(5\%\) and reduces the timing of each female runner by \(6\%\).
Find the probability that, after being equipped with the new apparel, the total time taken by \(2\) randomly chosen female runners differs from twice the time taken by a randomly selected male runner by less than \(17\) minutes.[4]