MSCI 212 Statistical Methods for Business 2019 EXAMINATIONS PART II
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2019 EXAMINATIONS
PART II (Second, Third and Final Year)
MANAGEMENT SCIENCE
MSCI 212 Statistical Methods for Business
Question 1
A basketball player attempts 20 shots from the field during a game. This player generally hits about 35% of these shots.
(a) In order to use a binomial model for the number of made baskets, what assumptions are needed in this example? (5 marks)
(b) How many baskets would you expect this player to make in the game? (5 marks)
(c) What is the probability that a player hits more than 11 shots (12, 13, 14, …, or 20)? (5 marks)
(d) How many points would you expect the player to score if all of these are 2-point shots? (5 marks)
(e) If this player randomly takes half of the shots from 3-point range and half from 2-point range and makes both with 35% chance, how many points would you expect the player to score? (5 marks)
Question 2
A car magazine journalist wants to investigate whether British or Japanese cars are generally more likely to break down. The table below shows the data collected by the journalist on the percentages of cars that broke down at least once a year across a sample of 18 car dealerships, 9 selling British cars and 9 selling Japanese cars. The editor of the car magazine has asked you to conduct appropriate statistical tests to compare the reliability of British and Japanese cars.
Percentages of British cars that broke down (%) |
9 |
5 |
10 |
4 |
14 |
30 |
22 |
22 |
12 |
Percentages of Japanese cars that broke down (%) |
4 |
11 |
10 |
20 |
7 |
30 |
18 |
14 |
11 |
(a) Under what conditions can we apply a parametric (i.e. z or t) test to compare the reliability of British and Japanese cars? (3 marks)
(b) Supposing that the conditions in part (a) above hold, conduct the appropriate parametric test at the 5% level of significance to compare the reliability of British and Japanese cars. Clearly state your null and alternative hypotheses and explain your conclusion. (15 marks)
(c) Supposing that the conditions in part (a) do not hold, recommend which non- parametric test would be appropriate to compare the reliability of British and Japanese cars. Justify your choice of test and clearly state the null and alternative hypotheses. (7 marks)
Question 3
Patients who are given a hospital appointment but do not turn up for it are a serious problem for the NHS in two important respects. They reduce the efficiency with which hospital clinics can be run and they can harm their own health. These patients are referred to as DNAs (Do Not Attends). Suppose that the target DNA rate for hospitals is 10%.
(a) Suppose that at a particular clinic that has 20 patients booked in for appointments, there is a probability of 0.1 that each patient is a DNA. Which probability distribution is most suitable to model the number of patients who attend a clinic? Justify your answer carefully. (5 marks)
(b) Two hospital doctors each have one clinic per week, with 20 patients booked in for each of them. Data from 100 such clinics held over a 12 month period is given below.
No. of DNAs |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 or more |
No. of clinics |
8 |
22 |
26 |
20 |
11 |
7 |
3 |
3 |
Use a goodness of fit test to show that they are unlikely to have come from a Binomial (n=20, p=0.1) distribution. (15marks)
(c) One possible explanation of why the above data does not come from a Binomial (n=20, p=0.1) distribution is: the DNA rates for the two doctors are different.
Explain carefully how you would investigate if this possible explanation is supported by the data. (You may assume that the data on the clinics for the two doctors can be separated.) (5 marks)
Question 4
The cost of treating individual surgical patients in hospital varies widely. The cost is believed to depend to some extent on the length of time that the patient stays in hospital and the seriousness of the operation they undergo. Data has been collected on the cost of treating 200 such patients and is summarised at the end of the question in the form of descriptive statistics, produced using SPSS. The cost of hospital treatment (in £s) has been regressed separately against their length of stay (in days) and against their operation duration (in minutes) using SPSS, with the results shown in the following pages under the headings MODEL 1 and MODEL 2 respectively.
(a) Which of these models show evidence of a significant linear relationship between treatment cost and the explanatory variable? Justify your answer. Express any significant relationship(s) in words. (5 marks)
(b) On the basis of the output provided, comment on the apparent quality of the two models. (4 marks)
(c) The results of using SPSS to predict the cost of treating a patient who stays in hospital for 4 days and has an operation duration of 55 minutes for the two models are shown below.
|
Cost (£) |
||||
Prediction |
LMCI |
UMCI |
LICI |
UICI |
|
MODEL 1 |
1042.97333 |
1022.35681 |
1063.58985 |
797.36504 |
1288.58161 |
MODEL 2 |
1246.70921 |
1220.05476 |
1273.36367 |
879.59543 |
1613.82299 |
Use your preferred model to advise the hospital management on the likely cost of an average patient with these characteristics, and of an individual patient with these characteristics. Comment on the level of accuracy of your predictions. (5 marks)
(d) Assuming that the model implied by your preferred regression is appropriate, sketch a diagram showing how the data would be spread in comparison to the regression line. Indicate the correct scales on your diagram, as far as possible. (7 marks)
(e) Explain in terms of the diagram you created in part (d) of this question what you would expect to see that would indicate that the errors in the model did not obey the regression assumption concerning constant variance. (4 marks)
2023-05-18