2012 ACJC P2 Q9

Each day a train travels from Town \(A\) to Town \(B\) and it is scheduled to arrive at Town \(B\) at \(\mathrm{0800}\) hrs. During the first year of operation it leaves Town \(A\) at \(0700\) hrs and is found to be late arriving at Town \(B\) on an average of once every \(4\) days. In the second year it leaves Town \(A\) at \(0650\) hrs and is late once every \(20\) days. Assuming the time of journey has a normal distribution, find, to the nearest minute, the standard deviation of the time of journey.[4]

Each day another train travels from Town \(B\) to Town \(C\). The train is scheduled to arrive at Town \(C\) at \(1200\) hrs. The time taken for this journey is a normal random variable with mean \(80\) mins and variance \(9\). Find the last time of departure of the train (to the nearest minute) from Town \(B\) which ensures that its probability of arriving late at Town \(C\) does not exceed \(0.02\).[2]

The departure time of the train from Town \(B\) is now fixed at \(1040\)hrs and the arrival time of the train in Town \(C\) is observed over \(20\) days. Find the probability that the average arrival time of the train is within \(2\) minutes of the scheduled arrival time of \(1200\) hrs.[3]