2023 NYJC P2 Q11

In this question, you should state the parameters of any distributions that you use.

The patients’ waiting times in minutes to consult Doctor Tan have the distribution \(N\left( 30,\ {{a}^{2}} \right)\) and the patients’ waiting times in minutes to consult Doctor Lim have the distribution \(N\left( 35,\ {{5}^{2}} \right)\).

1. The probability that the waiting time of a patient consulting Doctor Tan is between \(10\) to \(30\) minutes is \(0.49379\). Show that \(a=8\) to the nearest integer.[2]
2. Find the probability that the total waiting time of \(4\) randomly chosen patients consulting Doctor Tan is less than \(90\%\) of the total waiting time of \(3\) randomly chosen patients consulting Doctor Lim.[4]
3. State an assumption needed for your calculations in part (b) to be valid.[1]
4. Find the shortest waiting time, to the nearest minute, for Doctor Lim’s consultation to be exceeded by less than \(15\%\) of the patients waiting to see him.[3]
5. Ten patients waiting to see Doctor Lim are selected at random. Find the probability that more than \(4\) of these patients will have to wait for more than \(35\) minutes.[2]