5. A call centre records the length of time, \(T\) minutes, its customers wait before being connected to an agent. The random variable \(T\) has a cumulative distribution function given by
$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r }
0 & t < 0
0.3 t - 0.004 t ^ { 3 } & 0 \leqslant t \leqslant 5
1 & t > 5
\end{array} \right.$$
- Find the proportion of customers waiting more than 4 minutes to be connected to an agent.
- Given that a customer waits more than 2 minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes.
- Show that the upper quartile lies between 2.7 and 2.8 minutes.
- Find the mean length of time a customer waits to be connected to an agent.