2026 RI TP Q12

In this question you should assume that \(T\), \(X\) and \(Y\) follow independent normal distributions. You should also state the parameters of any distribution you use.

A customer service call centre categorises incoming calls for follow-up resolution of the issues. The time taken to categorise an incoming call, \(T\) minutes, follows a normal distribution \(\mathrm{N}(2, 0.2^2)\). Incoming calls are categorised as routine or complex calls. The time taken to resolve a routine call, \(X\) minutes, follows a normal distribution \(\mathrm{N}(5, k)\) and the time taken to resolve a complex call, \(Y\) minutes, follows a normal distribution \(\mathrm{N}(20, 6^2)\). Any incoming call is first categorised as either routine or complex and is immediately followed up with the resolution of the issue. The duration of the call is taken to be the sum of the times taken to categorise the call and to resolve the issue.

1. Sketch the distribution of \(T\) for \(1.2 \le t \le 2.8\).[2]

For a randomly chosen incoming call that is categorised as routine and then resolved, there is a probability of 0.254 that the duration of the call is more than 8 minutes.

2. Show that \(k = 2.24\) correct to 2 decimal places.[3]
3. On a weekday morning where there are 20 incoming calls, \(n\) calls are categorised as complex calls. Given that there is a probability of at most 0.01 that the mean time taken to resolve the \(n\) complex calls exceeds 24 mins, write down an inequality involving \(\sqrt{n}\) and hence find the possible values of \(n\).[3]

There is a review of the resolution processes such that the time taken to resolve a complex call is now reduced by 20%.

4. Find the probability that the time taken to resolve 2 complex calls is more than twice the time taken to resolve a routine call.[3]