a) Consider people’s preferences in terms of chocolate and their eye colour, presented as probability tree diagram below:

If 30% of people have brown eyes, 40% have green eyes and the remaining people have blue eyes, and it is found that 70% prefer milk chocolate , 10% prefer white chocolate and the remaining prefer plain chocolate, using the tree diagram above.

Required:
Calculate the probability that a person:
i) P ( Prefer plain Chocolate and has Brown Eyes)                                                                                               (1.5 marks)

ii) P ( Prefer milk Chocolate and has Brown Eyes)                                                                                               (1.5 marks)

iii) P ( Prefer white Chocolate and has Brown Eyes)                                                                                             (1.5 marks)

iv) P ( Prefer plain Chocolate and has Green Eyes)                                                                                               (1.5 marks)

v) P ( Prefer milk Chocolate and has Green Eyes)                                                                                                 (1.5 marks)

vi) P ( Prefer white Chocolate and has Green Eyes)                                                                                               (1.5 marks)

vii) P ( Prefer plain Chocolate and has Blue Eyes)                                                                                                 (1.5 marks)

viii) P ( Prefer milk Chocolate and has Blue Eyes)                                                                                                (1.5 marks)

ix) P ( Prefer white Chocolate and has Blue Eyes)                                                                                                 (1.5 marks)

b) If three fair coins are tossed,
Required:
Determine, the probabilities of
i) getting two heads and a tail                                                                                                                                  (3.5 marks)

ii) getting three tails                                                                                                                                                 (3 marks)

The mileages recorded for a sample of company vehicles during a given week gave the following data:

Required:
a) Draw the frequency table of this data given a class size of 5.                                                                                         (6 marks)

b) Determine mean, median and mode for the data and comment on the distribution of data.                                           (6 marks)

c) Compute the quartile deviation.                                                                                                                                      (4 marks)

d) Determine the standard deviation for the data.                                                                                                               (4 marks)

a) A particular linear programming problem is formulated as follows:
Minimize z = 2500x + 3500y
Subject to constraint:

Required:
i) Draw these constraints on the same graph paper.                                                                                                           (6 marks)

ii) Determine the optimum solution.                                                                                                                                  (6 marks)

b) Managers within a subsidiary company in a conglomerate want to know how to maximize the profit from two types of products X and Y. Each product X requires one hour of labour and six litres of molding material, whereas each product Y requires two hours of labour and five litres of molding material. The total labour hours available for each week is 40 and the total amount of molding material each week is 150. The profit contribution from product X is GH¢20 and from product Y is GH¢ 30.
Required:
Formulate, the linear programming model for the problem.                                                                                              (8 marks)

a) The AXM manufacturing company has determined that the cost function for producing a particular type of pavement block is given by:

where x is measured in number of units and C(x) in GH¢.

Required:
i) Calculate the derivative of C(x) with respect to x .                                                                                                          (5 marks)

ii) Determine how quickly the cost is changing at x =1000.                                                                                               (4 marks)

b) At the Zee manufacturing company, the marginal cost for producing x gears, measured in hundreds is

MC =10+0.1x .
If the fixed cost (the cost of producing zero items) is GH¢3000.
Required:
Determine the cost for manufacturing 5000 gears.                                                                                                            (11 marks)

Required:
Determine the most preferred business opportunity if:

[ NB: You are required to discount these future sums by using the interest rate of 8%. ]

i) GH¢7000.00 is paid in 4 years’ time.                                                                                                                             (7 marks)

ii) GH¢8000.00 is paid in 6 years’ time.                                                                                                                             (7 marks)

Calculate the accumulated amount if GH¢2500.00 were invested at 18% compound interest for a period of six years.        (6 marks)

a) Given the cost function

Required:
i) Evaluate C(+2) ,                                                                                         (2.5 marks)

ii) Evaluate C(-2) .                                                                                         (2.5 marks)

b) Evaluate each of the following for the given value:

 

c) A paper producing company has determined that its profit from selling x hundred boxes of envelopes is given by the expression:

Required:
i) Determine the number of boxes the company must sell to break even.                                                                          (5 marks)

ii) Determine the number of boxes the company must sell to make money.                                                                      (5 marks)

The personnel department of Dropper Ltd, a large cocoa processing company in DropperLand, is concerned about absenteeism among its shop floor workforce. There is a general feeling that the underlying trend has been rising, but nobody has yet analysed the figures. The total number of shops employees has remained virtually unchanged over the few years.
The mean number of absentees per day for each quarter of the years 1999 to 2001 and quarter 1 in 2002 is given in the table below:

Required:
a) Plot the data on a graph, leaving space for the remaining 2002 figures.                                                                        (3 marks)

b) Using the method of 2 quarterly centered moving,
i) determine the trend in the series and superimpose this on your graph in (a) above.                                                       (4 marks)

ii) determine the equation of the trend line above by considering only the first and last centered moving average value on your graph in (i) above.                                                                                                                                                                (3 marks)

c) Using an appropriate decomposition model to determine the seasonal variations in the data. Give reasons for your choice of model.                                                                                                                                                                                  (5 marks)

d) Use your analysis above to produce roughly forecasts of the mean number of absentees there will be in the remaining quarters of 2002. Comment on your forecast.                                                                                                                                   (5 marks)

a) Sketch Graph(s) to show the following critical points on a cost or revenue function:
i) Local Maximum Point                                                                    (1 mark)
ii) Absolute Maximum Point                                                             (1 mark)
iii) Local Minimum Point                                                                  (1 mark)
iv) Absolute Minimum Point                                                             (1 mark)
v) Point of inflexion                                                                           (2 marks)

b)

If the fixed market price of low grade steel is GH¢ 6.90 and the fixed market price of high grade steel is GH¢ 13.80.
Required:
i) Determine the number tons of low grade steel and high grade steel to be produced to maximize total revenue.            (10 marks)

ii) Determine the maximum total revenue.                                                                                                                         (4 marks)

Suppose that Mr. Kuu, a retired chartered accountant, is facing a decision about where to invest that small fortune that remains after he has deducted the anticipated expenses for the next year from the earnings from his consultancy job. An investment analyst has suggested to him two types of investment, and to help make the decision he obtained some rates of return from each type. He would like to know what he can expect by way of the return on his investment, as well as other types of information, such as whether the rates are spread out over a wide range (making the investment risky) or are grouped tightly together (indicating a relatively low risk). The returns for the two types of investments are listed here.

Required:
a) Draw histograms for each set of returns.                                                                                                                        (5 marks)

b) Compute the following measures of central tendency for the set of returns:
i) Mean                                                                                                                                                                                (3 marks)

ii) Median                                                                                                                                                                           (3 marks)

iii) Mode                                                                                                                                                                             (3 marks)

c) Compute the following measures of spread for the set of returns:
i) Standard deviation                                                                                                                                                            (3 marks)

ii) Coefficient of variation                                                                                                                                                   (2 marks)

d) Using (a), (b) and (c) above, which investment should Mr.Kuu choose and why?                                                         (1 mark)