### Explain briefly why it might be reasonable to expect these calls to arrive at random

**Question 1**

The call centre of a major US energy provider (USGEN) expects to be particularly busy during the early shift in the early Winter. During the early shift calls arrive at a mean rate of 180 per hour and are believed to arrive *at* *random*.

- (i) Explain briefly why it might be reasonable to expect these calls to arrive
*at**random*. - (ii) What would be the probability distribution of the number of calls arriving during a five-minute period in the early shift? (There is no need to calculate any probabilities in this part of the question).
- (iii) Show (using probability tables or probability formulae) that the probability that there are more than 20 calls in a five-minute period is 0.083. Show your working.
- (iv) Staffing levels during the early shift are such that they can cope with occasional peaks in arrivals, but service levels deteriorate rapidly when they experience a number of peaks close together. Continuing to assume that calls arrive at random, show that the probability that there are 6 or more five-minute periods in an hour in which the number of calls exceeds 20 is less than 1 in 1000. Show your working.
- (v) Quiet periods only occur very rarely during the early shift in the USGEN call centre, so when gaps between calls exceed 2 minutes the management takes it as a signal of a telephone system failure and resets the system. What is the chance they reset the system unnecessarily in response to a 2 minute gap? Justify your method.

- (vi) You have seen in class how a Normal distribution can be used to approximate a Binomial distribution under certain conditions on n and p. (Remember that the Normal distribution was chosen to match the Binomial distribution in mean and standard deviation A Normal distribution can also be used to approximate a Poisson distribution under certain conditions. By applying the same idea, use the Normal distribution to provide an approximate answer to part (iii) above. Explain your method carefully.
- (vii) Suggest the conditions under which a Normal distribution can be used to approximate a Poisson distribution. Justify your answer.

**Question 2**

As at workshops 2 and 4, use <Transform><Random Number Generators> to set your unique starting point for the SPSS random number generator. For this coursework question use the last four digits of your library card **PLUS 1**, i.e. if your library card ends ‘4321’ type in ‘4322’, and if your library card ends ‘4329’ type in ‘4330’. **RECORD THESE FOUR DIGITS AT THE TOP OF YOUR ANSWER.**

*Continued on next page*

Accident and Emergency Departments (AEDs) in hospitals in England have to deal with highly variable workloads, providing high quality healthcare within a reasonable length of time. Data gathered on 183 organisations which provide accident and emergency (A&E) services in England is contained in the SPSS data file MNGT212_AED_Performance.sav. The data relates A&E attendances during June 2014 and contains the following variables:

- Name – name of organisation providing the A&E service;
- Attendances – number of attendances during June 2014;
- Left – number of patients who left before being seen for treatment;
- Reattendances – number of patients who reattended within 7 days.
- TtoT_median –median time before treatment started (minutes);
- TtoT_95percentile – 95th percentile of time before treatment started (mins);
- TimeinAE_median – median time in A&E (mins);
- TimeinAE_95percentile – 95th percentile of time in A&E (mins);
- PercentReatt – percentage of attendances which were reattendances;
- PercentLeft – percentage of patients who left before being seen for treatment.

The government monitors the performance of AEDs in terms of:

- times before patients receive treatment (or are reassured that the do not need treatment);
- total time that patients spend in A&E – e.g. one of the government targets is that 95% of patients should leave the AED within 4 hours;
- percentage of repeat attendances;
- percentage of patients who leave before a treatment decision has been reached.

Suppose that one of the hospital organisations (Great Western Hospitals NHS Trust – GWHNHST) has recently been criticised in the local media about the time its patients spend in A&E. In particular the media have noted that its 95th percentile point of the time in A&E is 298 mins, i.e. considerably higher than the government target of 240 mins. You have been asked by GWHNHST management to investigate the available data further. Suppose that you only have time to obtain data from a random sample of 75 of the remaining 182 hospital organisations. Draw a random sample of 75 organisations from the other 182 organisations and investigate your sample using SPSS.

- (a) In no more than 6 pages describe the main features of your sample as if to the GWHNHST management. You should include main features of individual variables and interesting relationships between them. You may include SPSS numerical and graphical output and/or you may quote values from your SPSS output. (The clarity and content of your report are both important). [Worth about 75% of the marks]

- (b) Using your sample as evidence, provide your own view on the performance of GWHNHST and what scope there might be for improving performance. [No more than 2 pages. Worth about 25% of the marks]