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MNGT 212: Statistical for Business Methods

Coursework Resit Paper 2015

Answer all five questions and follow submission instructions on page 15.

Question 1

A major manufacturer provides spare parts for one of its products. Orders received on each weekday are delivered overnight in a specially designed vehicle. Orders are always for one item and are almost always in response to a breakdown in the customer’s equipment. The vehicle has a capacity of 3 items, and any items which cannot fit into the vehicle are delivered by a special courier at considerable extra cost.

(i)Assuming that each day there is a 10% chance that the vehicle capacity is exceeded, what is the probability distribution of the number of delivery trips over the 5-day week for which a special courier is required? Justify your answer carefully, stating any extra assumptions that you need to make.

(15% of marks)

(ii)What is the probability that the special courier will be required on at least two days in a 5-day week?

(15% of marks)

(iii)Suppose you are told that the average number of items ordered per day is 1.5. Explain briefly why it might be reasonable to assume that the number of items ordered on a weekday has a Poisson (1.5) distribution, and use this distribution to confirm or revise the chance that the vehicle capacity is exceeded each day.

(30% of marks)

(iv)The manufacturer is considering whether to increase its vehicle capacity to 5 instead of 3. Using the Poisson (1.5) distribution for the daily number of items ordered, compare the probabilities that no more than one special delivery will be required over a 2 week period for the two sizes of vehicle.

(40% of marks)

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Question 2

A theory has been suggested that firms most likely to attract take-over bids are those achieving relatively poor financial returns. One measure of such performance is through ‘abnormal returns’, which average zero over the whole population and for which a negative value indicates a poor return. The ‘abnormal returns’ for a random sample of 81 firms for which take-over bids had been made were analysed and had a mean of -0.0029 and a standard deviation of 0.0169 in the 1-year period prior to the take-over bids occurring.

(i)Calculate a 95% confidence interval for the mean ‘abnormal return’ for companies for which take-over bids are made. Justify your choice of calculation method.

(25% of marks)

(ii)Propose and carry out a test of the above theory, clearly stating the null and alternative hypotheses. Use a 5% significance level. State your conclusion clearly in layman’s (i.e. non-statistical) terms.

(30% of marks)

(iii)If the same mean and standard deviation had been obtained from a sample of 17 (instead of 81), retest your hypotheses. State any additional assumptions that you have made.

(20% of marks)

(iv)Explain briefly the theoretical advantages of having a sample of size 81 rather than a sample of size 17. Do your analyses and results in (ii) and (iii) demonstrate these advantages?

(25% of marks)

Question 3

Given below are the earnings per share for 1987 and 1988 for a sample of 10 corporations (Source: Business Week, 1989 Bonus Issue):

Corporation

1987

1988

IBM

8.72

9.27

Sears-Roebuck

4.35

2.72

Chevron

3.65

5.17

Walt Disney

2.85

3.80

American Express

1.20

2.31

McDonalds

2.89

3.43

Anheuser-Busch

2.04

2.45

Kellogg

3.20

3.90

JC Penny

4.11

6.02

Motorola

2.39

3.43

Question 3 continued overleaf

(i)The SPSS outputs from two parametric and two non-parametric analyses of this data are shown below and overleaf. Choosing the most appropriate test, is there any evidence that mean earnings per share were higher in 1988? Clearly state your null and alternative hypotheses, justify your choice of test and state your conclusions.

(35% of marks)

(ii)For each of the tests above that you have NOT used in part (a) describe changes to the details of the problem which would make that test the most appropriate. What would your conclusion be for each test?

(35% of marks)

(iii)Suppose that the first 5 companies have a characteristic in common that is absent for the other 5 companies. Explain carefully how you would test whether the five companies with the characteristic have experienced bigger increases in earnings than the other five companies. Justify your choice of test. [There is no need to specify any SPSS commands.]

(30% of marks)

Independent Samples T-Test

Paired Sample T-Test

Question 3 continued overleaf

Mann-Whitney Test

Wilcoxon Signed Ranks Test

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Question 4

A cross-channel ferry company is interested in predicting the amount of duty-free goods (e.g. wines, spirits, beer, cigarettes, perfumes) that it can expect to sell to passengers as they cross the channel. The following data has been collected for 100 crossings:

·sales: value of sales of duty-free (£’s);

·adults: number of adult passengers;

·children: number of child passengers;

·allpass: total numbers of passengers.

Descriptive statistics produced in SPSS are:

The value of sales has been regressed against each of the passenger variables separately using SPSS, with the results shown at the end of the question under the headings MODEL 1, MODEL 2 and MODEL 3.

a) Which of these three sets of results show evidence of a significant linear relationship between sales and the explanatory variable? Justify your answer.

(15% of marks)

b) Compare models 1 and 2 on the basis of the output provided. Which of the two models do you think is better on statistical grounds? Justify your answer and suggest a practical reason why this might be so.

(15% of marks)

c) Compare your preferred model from part (b) with model 3. Which of the two models do you think is better on statistical grounds? Justify your answer and suggest a practical reason why this might be so.

(15% of marks)

d) The results of using SPSS to make predictions using each model are shown overleaf for (i) a crossing carrying 300 adults and 100 children, and (ii) a crossing carrying 450 adults and 250 children. Using your preferred model from part (c), use the 95% CIs for mean and/or individual values to advise the ferry company about the level of sales of duty free goods to expect on crossings with these numbers of passengers. Include comments on the accuracy and any other limitations of your advice.

(30% of marks)

Question 4 continued overleaf

MODEL

Crossing

Prediction

LMCI

UMCI

LICI

UICI

1

(i)

3880.8

3764.2

3997.5

2792.7

4969.0

1

(ii)

5627.6

5359.5

5895.8

4513.0

6742.2

2

(i)

3629.0

3434.2

3823.9

1723.9

5534.2

2

(ii)

4984.1

4416.1

5552.2

3005.6

6962.6

3

(i)

3845.3

3711.5

3979.1

2590.0

5100.5

3

(ii)

6061.2

5679.0

6443.4

4755.9

7366.5

e) One of the more thoughtful of the ferry company managers is worried about model 2. It seems to imply that each child on the ferry will spend, on average, about £9 on duty-free goods. However children are not allowed to buy duty-free goods! Whether or not you have recommended model 2, the manager is starting to lose faith in the idea of using regression models, given that they seem to be capable of producing nonsensical results. Explain, as if to the manager, why model 2 could be providing these apparently silly results.

(25% of marks)

MODEL 1: Regression Analysis

Question 4 continued overleaf

MODEL 2: Regression Analysis

Question 4 continued overleaf

MODEL 3: Regression Analysis

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Question 5

There has been considerable concern about high attendance rates at NHS accident and emergency departments (AEDs) this last Winter. In an attempt to understand some of the drivers of the attendance rates, data has been collected on attendances to one AED during the Winter from 50 electoral wards (geographical areas). For each electoral ward, data is also available on the population size and age structure, the level of social deprivation, and the distance to the AED.

It is well known that attendance rates depend on a person’s age, so for each electoral ward an age-standardised attendance rate (ASAR) has been calculated. For example a ward with an ASAR of 1.2 would have 20% more attendances than would be expected given the age structure of its population; and a ward with an ASAR of 0.9 would have 10% less attendances than would be expected given the age structure of its population.

Senior NHS management would like to know whether these ASARs also depend on social conditions or on the distance that wards are from an AED. For each ward the following data has therefore been obtained:

·ASAR – age-standardised attendance rate;

·depscore – deprivation score – increasing as level of social deprivation increases (0 –10);

·distance – distance between electoral ward and AED (miles).

Answer the following questions using the SPSS outputs (provided at the end of the question) as you think appropriate.

a)Comment briefly on the SCATTERPLOTS and CORRELATIONS. What signs are there of interesting linear or non-linear relationships?

(20% of marks)

b)Using the results of running STEPWISE regression analyses (starting with no variables), see STEPWISE (Starting with no variables) I, state clearly the regression model that this analysis is recommending, including details of the error term. Interpret the coefficients of the model and the details of the error term, as if to an AED manager.

(35% of marks)

c)Comment briefly on each element (i.e. 3 scatterplots and the tests of Normality) of the RESIDUALS ANALYSIS I. What evidence is there of a breakdown in the usual regression assumptions?

(20% of marks)

d)Someone suggests that a better regression model might be achieved using ‘reciprocal of distance’ (i.e. 1/distance) as an additional variable, labelled ‘recipDist’. Using the results of running STEPWISE regression analyses (starting with no variables), see STEPWISE (Starting with no variables) II, and the associated residuals analyses, see RESIDUALS ANALYSIS II, explain carefully what this new analysis seems to imply. What model do you now recommend?

(25% of marks)

Question 5 continued overleaf

SCATTERPLOTS

CORRELATIONS

Question 5 continued overleaf

STEPWISE (Starting with no variables) I

Question 5 continued overleaf

RESIDUALS ANALYSES I

Question 5 continued overleaf

STEPWISE (Starting with no variables) II

Question 5 continued overleaf

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