MSCI212 Statistical Methods for Business 2021 EXAMINATIONS
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2021 EXAMINATIONS
PART II (Second, Third and Final Year)
MANAGEMENT SCIENCE
MSCI212 Statistical Methods for Business
Question 1
It is estimated that during any one hour period, an average of 10 Internet users visit the website of Sport-Equip Ltd, a company that sells sports equipment. Some of the users who visit the website end up buying sports equipment from the website while others are simply browsing in order to obtain product information.
(a) Clearly explain and justify which probability distribution you would use to describe the number of Internet users who visit the website of Sport-Equip Ltd in a one hour period. [There is no need to calculate any probabilities for this part of the question]
(5 marks)
(b) What is the probability that during any half-hour period, there will be less than 3 visitors to the website?
(5 marks)
(c) What is the probability that during any two-hour period, there will be more than
15 visitors to the website?
(5 marks)
(d) If a user has just visited the website, find the probability that the website will have another visitor within the next 10 minutes. In your answer, state the probability distribution you have used and explain your choice.
(4 marks)
(e) It is estimated that 40% of Internet users who visit Sport-Equip Ltd’s website buy a product from the company. If 100 users visit the website over a given period of time, find the probability that more than 50 of them will buy a product from the company. In your answer, state the probability distribution you have used and explain your choice.
(6 marks)
Question 2
Prior to a recent marketing campaign launched by one of its competitors, the daily sales of bread at a particular supermarket have been observed to follow a near Normal distribution with a mean of 400 loaves and a standard deviation of 40 loaves. In the 35 days since the marketing campaign the mean daily sales has been 380 loaves.
(a) Propose and carry out a test to see whether there has been a decrease in the daily sales of bread since the competitor’s marketing campaign, clearly stating your null and alternative hypotheses. Use a 5% significance level. Justify your choice of test and state your conclusion clearly.
(6 marks)
(b) Using the statistical test in part (a), what is the chance that the wrong conclusion is reached if the mean daily sales of bread has not changed since the marketing campaign? Justify your answer.
(2 marks)
(c) Calculate the power of the statistical test used in part (a) if the true mean daily sales after the marketing campaign were to drop by 10%. Sketch out the shape of the power curve marking those points known to you.
(8 marks)
(d) When the source of the data described above is looked at more closely, it turns out that the values of mean and standard deviation prior to the marketing campaign were in fact only calculated from the previous 3 month period, i.e. 90 days, and for the 35 days since the campaign the standard deviation of loaves sold was 44. Revise your proposed test accordingly and re-test for a decrease in sales. Justify your choice of test and state clearly any additional assumptions that you have made.
(9 marks)
Question 3
Nationally 60% of trainee accountants taking a particular professional accountancy examination pass the examination.
(a) One particular examiner is given the scripts of a random sample of 10 trainees to mark. What is the probability distribution of the number of trainees passing the examination? Justify your answer carefully.
(6 marks)
(b) The number of trainees who passed at each of a random sample of 80 training centres, each of which entered 10 trainees for the exam is given below.
No. of passes |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
No. of Centres |
1 |
3 |
5 |
7 |
9 |
12 |
17 |
12 |
7 |
3 |
4 |
Use a goodness of fit test to show that these data are unlikely to have come from a Binomial (n=10, p=0.6) distribution.
(13 marks)
(c) Explain carefully why the data described in (b) might not come from a Binomial (n=10, p=0.6) distribution, making reference to likely relevant characteristics of real training centres.
(6 marks)
Question 4
The managers of a theme park believe that daily sales from fast-food and ice-cream outlets depend, to some extent, on the numbers of visitors entering the park each day. Data have therefore been collected for 250 days, and consist of:
fastfood – daily sales from fast-food outlets (£)
icecream – daily sales from ice-cream outlets (£)
visitors – number of people visiting the park.
The basic descriptive statistics for these variables are given below.
Descriptive Statistics
|
N |
Minimum |
Maximum |
Mean |
Std. Deviation |
FASTFOOD ICECREAM VISITORS Valid N (listwise) |
250 250 250 250 |
2833.00 2023.00 3715.00 |
83187.00 56025.00 21384.00 |
30079.27 21022.05 12118.32 |
18233.80 12403.64 3948.3332 |
The value of sales of each of the two types of food have been regressed separately against the numbers of visitors entering the park using SPSS, with the results shown at the end of the question under the headings MODEL 1 and MODEL 2 respectively.
(a) Which of these results show evidence of a significant linear relationship between food sales and number of customers entering the park? Justify your answer.
(4 marks)
(b) On the basis of the output provided, comment on the apparent quality of the two models.
(4 marks)
(c) The results of asking SPSS to predict the sales of the two types of food using models 1 and 2 respectively on a day when there are 10,000 visitors are shown in the table below. [Note that the row containing results for ‘Model 3’ is not needed until section (d)].
|
prediction |
lmci |
umci |
lici |
Umci |
Model 1 |
23755.51 |
21784.04 |
25726.97 |
-3770.91 |
51281.93 |
Model 2 |
17808.20 |
16269.05 |
19347.36 |
-3682.11 |
39298.51 |
Model 3 |
41563.71 |
40807.17 |
42320.26 |
31000.53 |
52126.89 |
Advise the theme park management on the level of fast-food and ice-cream sales that they should plan for both on an average day with 10,000 visitors and on an individual day with 10,000 visitors. Comment on the level of accuracy of your predictions.
(5 marks)
(d) In a moment of inspiration the values of the two types of food sales are added together to give a new variable, called ‘ffandic’ . The results of regressing this new variable against number of visitors are shown below as MODEL 3. SPSS’s predictions for a day with 10,000 visitors are shown in section (c). Compare this third model and its predictions with those from your previous two models, identifying what you think would be likely to be its main advantages and disadvantages for management.
(7 marks)
(e) Suggest and explain some practical circumstances that might give rise to the observed relative performance of the three models.
(5 marks)
2022-06-09