2018 SRJC MYE P2 Q12

In a polyclinic, the waiting time, in minutes, for dentist \(A\) and dentist \(B\), are modelled as having normal distributions with means and standard deviations as shown in the table below.

1. Find the probability that on a randomly chosen day, the waiting time for a randomly chosen patient waiting to see dentist \(B\) differs from twice the waiting time for a randomly chosen patient waiting to see dentist \(A\) by more than \(5\) minutes.[3]
2. Two randomly chosen patients who were each queueing to see dentist \(A\) and one randomly chosen patient who was queueing to see dentist \(B\) each asked for their waiting time. Find the probability that the average waiting time for these three patients to see their dentists is at most half an hour.[3]
3. State an assumption needed for your calculations in (i) and (ii).[1]

The random variable \(V\) denotes the waiting time, in minutes, for a patient to make payment right after the consultation with any dentist in the polyclinic. The random variable \(V\) follows a normal distribution with mean \(\mu \) and standard deviation \(\sigma \).

4. If \(\mathrm{P}\left( \mu -4 < V < \mu -\frac{1}{2} \right)=2\mathrm{P}\left( \mu -\frac{1}{2} < V < \mu +\frac{1}{2} \right)\) and \(\mathrm{P}\left( \mu -4 < V < \mu +\frac{1}{2} \right)=\mathrm{P}\left( V>\mu \right)\), find the value of \(\sigma \).[3]

Peter started waiting to make payment at 11.30 am.

5. Hence find the range of values of \(\mu \) if the chance for him to leave the polyclinic after 11.35 am is at least \(90\%\).[3]