Explain in detail.

(a) It is assumed that the number of calls received at a call centre each minute follows a Poisson distribution with mean 3. The call centre operates for 12 hours a day from 8am to 8pm. (i) Explain why it is reasonable to assume that the number of calls received in a minute follows a Poisson distribution. [2] (ii) Find the probability that there will be less than 4 calls in a randomly chosen minute. [3] (iii) What is the probability that there will be 5 or more calls in a randomly chosen 2 minute period? [3] (iv) How many calls would you expect the call centre to receive during the operating period each day? I [2] (v) The call centre will be extending operations to cover a 24 hour period. Can the call centre still assume that the number of calls each minute follows a Poisson (3) distribution? Justify your answer. [3] (b) The call centre has a target of completing 90% of calls within 7 minutes. The time taken to complete a call is the time the customer spends talking to an agent. A random sample of 200 calls were taken over the period of one day. 23 of these calls exceeded 7 minutes. Let P be the sample statistic of the proportion of calls that are completed within 7 minutes. (i) Assume that the population proportion, p = 0.90. Write down the approxi- mate sampling distribution of P for samples of the same size as the experiment conducted here. [3] (ii) Construct a 95% confidence interval for the proportion of calls that are com- pleted within 5 minutes. [6] (iii) On the basis of the statistical evidence from this random sample, is the call centre meeting its target? Justify your answer. [3]