2014 NJC CT Q9

The normal random variable \(X\) has mean \(\mu \) and variance \({{\sigma }^{2}}\). It is known that \(\mathrm{P}\left( \left| X-\mu \right|>7.5 \right)=0.2113\). Calculate the value of \(\sigma \).

Earl, a student in Reverie Junior College, commutes to school by a particular public bus service. On any one school day, the amount of time he has to wait before the particular bus arrives is normally distributed with mean \(16\) minutes and standard deviation \(5\) minutes. Upon boarding the bus, the amount of time the bus takes to reach his school is normally distributed with mean \(28\) minutes and standard deviation \(9\) minutes. Earl is late for school if he reaches after \(7.35\)am.

Suppose on a randomly chosen school day, Earl starts to wait for the bus at \(6.40\) am. Find the probability that Earl is late on that day.

After getting a yellow slip for being late a few times, Earl decides to leave his house earlier. Find the latest time he needs to start waiting for the bus such that the probability that he is late does not exceed \(5\%\).