2026 May TZC P2 Q8

The waiting time, \(T\) minutes, of patients at a clinic can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The waiting times of all patients are independent of each other.

\( 3.8\%\) of the patients have a waiting time of less than \(20\) minutes, while \(7.3\%\) of the patients have a waiting time of more than \(30\) minutes.

1. 1. Find the value of \(\mu\) and the value of \(\sigma\).

   2. Hence, find the probability that a patient chosen at random has a waiting time of between \(15\) and \(23\) minutes.

      [7]

The doctor at the clinic only sees patients with an appointment.
Records show that \(82.6\%\) of patients attend their appointment.
The probability of a patient attending their appointment is independent of any other patient.

2. On a particular day, \(20\) patients each made an appointment to see the doctor.

   1. Find the probability that at least \(13\) of these patients attended their appointment.

   2. Given that at least \(13\) of these patients attended their appointment, find the probability that more than \(2\) patients did not attend their appointment.

      [7]