Problem

A company is considering a change to the way it deals with phone calls from its customers. The current average time taken to deal with a phone call is 6 minutes.

A random sample of 50 phone calls that have been dealt with using the current method is taken and the mean of this sample is found to be 6.4 minutes.

The mean deal time for calls dealt with using the current method, measured in minutes, can be modelled by a Normal distribution with standard deviation 1.2 minutes

(a) Carry out a suitable test, at the 5% level of significance, to investigate whether the mean deal time for calls dealt with using the current method is greater than 6 minutes

Give a clear conclusion to this test.

After further investigation the company decides to change the way it deals with phone calls from its customers.

The new deal time for a call, $W$ minutes, has probability density function

$f(w) = \begin{cases} \frac{1}{9}(6-w)(w-2) & 2 \leqslant w \leqslant 5 \\ 0 & \text{otherwise} \end{cases}$

(b) Find the mode of $W$

(c) Find $E(W)$

(d) Find the median deal time for calls dealt with using the new method.

The new deal time can be modelled by a Normal distribution with mean 4 minutes and standard deviation 0.935 minutes

(e) Using this model, find the probability that it will take more than 18 minutes to deal with 3 randomly selected calls dealt with using the new method.

(Total for Question 6 is 19 marks)