The waiting time \(T\), in minutes, for a patient at a medical clinic is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is known that \(3.8\%\) of patients wait less than \(20\) minutes and \(7.3\%\) wait more than \(30\) minutes.
[7]
Find \(\mu\) and \(\sigma\).
Hence find the probability that a patient waits between \(15\) and \(22\) minutes.
A doctor only sees patients who have made appointments. Records show that \(\frac56\) of patients who make appointments attend the clinic. Attendance by different patients is independent.
On a particular day, \(20\) patients have made appointments.[7]
Find the probability that at least \(13\) patients attend.
Given that at least \(13\) patients attend, find the probability that more than two patients do not attend.
The doctor can see at most \(40\) patients in one day. The clinic overbooks appointments and wants the probability that the doctor can see every patient who attends to be at least \(0.99\).
Find the greatest number of appointments the clinic should accept for one day.[4]