**WEDNESDAY: 7 December 2022. Morning Paper. Time Allowed: 3 hours.**

**Answer any FIVE questions. ALL questions carry equal marks. Show ALL your workings. Do NOT write anything on this paper.**

**QUESTION ONE**

1. State FOUR characteristics of a good measure of dispersion. (4 marks)

2. Explain the meaning of the following terms as used in set theory:

Venn diagram. (2 marks)

Complement of a set. (2 marks)

Union of a set. (2 marks)

3. The management team of Stage Supermarket is interested in determining whether there is any relationship between

the distance from customers’ residences to the supermarket and the number of visits made to the supermarket in a

particular period.

The following results were obtained:

**Required:**

Determine whether there is any relationship between the number of visits made to the supermarket and the distance from customers’ residences to the supermarket. (8 marks)

Calculate the decrease in the number of visits to the supermarket with each unit increase in distance of customers’ residences from the supermarket. (2 marks)

**(Total: 20 marks)**

**QUESTION TWO**

1. Shoetec Ltd., a manufacturer of stylish shoes, estimates that at full scale production, it would sell between 2,000 and 3,000 pairs of shoes.

The total monthly revenue in thousands of shillings over this range is represented by the function

TR = 3×2 + 7x.

The firm estimates that the marginal cost (MC) in thousands of shillings could be represented by the function

MC = 5×2 – 3x – 2 and fixed cost (FC) will be Sh.1,000 per month.

Where x is the monthly output in thousands of pairs of shoes.

**Required:**

Derive the average cost and average revenue functions of the firm. (4 marks)

Calculate the profit maximising output. (4 marks)

Calculate the price charged upon maximising profit and how much each pair of shoes would cost. (2 marks)

2. An examination was administered to a group of students and the results were as summarised below:

A candidate fails the examination if he/she obtains less than 40% in the examination. In order to pass with distinction, the candidate must obtain at least 75% in the examination.

**Required:**

Calculate the mean and standard deviation of the distribution of marks assuming that the marks scored are normally

distributed. (10 marks)

**(Total: 20 marks)**

**QUESTION THREE**

1. Distinguish between the following terms as used in probability:

“Conditional probability” and “marginal probability”. (4 marks)

“Discrete probability distributions” and “continuous probability distributions”. (4 marks)

2. The banking industry in a given country is controlled by three banks Faida Bank, Akiba Bank and Ahadi Bank. As

at 31 December 2020, Faida Bank controlled 30% of the market share, Akiba Bank controlled 50% of the market share and Ahadi Bank controlled 20% of the market share.

A survey was conducted on the market shares of the three banks as at 31 December 2021 and revealed the following:

1. Faida Bank had retained 80% of its market share and lost 15% and 5% to Akiba Bank and Ahadi Bank respectively.

2. Akiba Bank had lost 10% and 20% of its market share to Faida Bank and Ahadi Bank respectively.

3. Ahadi Bank had lost 5% of its market share to Faida Bank and 5% to Akiba Bank.

4. There were no significant changes in the banking habits of the customers during the year ended 31 December 2021.

**Required:**

Determine the transition matrix from the above information. (2 marks)

Determine the market shares of the three banks as at 31 December 2021. (3 marks)

The steady state market shares of the three banks. (7 marks)

**(Total: 20 marks)**

**QUESTION FOUR**

1. The Production Manager of Mechtex Ltd., a manufacturer of machines, is investigating a claim by customers about machine A and machine B that it manufactures.

The claim is that machine A has a longer useful life than machine B.

A sample of 60 machine As taken from the market reveals that the machine has a mean useful life of 28,000 hours

with a standard deviation of 900 hours. A sample of 80 machine Bs has a mean useful life of 30,000 hours with a

standard deviation of 1,000 hours.

**Required:**

Advise the Production Manager of Mechtex Ltd. if there is a significant difference in the useful lives of the machines. (8 marks)

Use a significance level of 5%.

2. The following data relate to the number of computers sold each day over the last 240 working days by a leading computer firm.

**Required:**

The modal number of computers sold. (2 marks)

The quartile deviation of the number of computers sold. (6 marks)

The quartile coefficient of skewness of the number of computers sold. Interpret your results. (4 marks)

**(Total: 20 marks)**

**QUESTION FIVE**

1. Highlight FOUR advantages of decision tree analysis as a tool for decision making. (4 marks)

2. State FOUR characteristics of the binominal distribution. (4 marks)

3. The management of a wall paint manufacturing company is faced with the problem of choosing one of three products

to add to its existing product line. The potential demand for each product may turn out to be good, moderate or poor

with probabilities estimated as 0.75, 0.15 and 0.10 respectively.

The estimated profit or loss under the three states of demand with respect to each product is outlined below:

**Required:**

Advise the management on the choice of product based on the expected monetary valve (EMV) criterion. (4 marks)

Compute the expected opportunity loss for each decision.

Which decision would you recommend based on the expected opportunity loss? (4 marks)

Compute the expected value of perfect information. (4 marks)

**(Total: 20 marks)**

**QUESTION SIX**

1. Explain THREE roles of quantitative analysis in the decision making of organisations. (6 marks)

2. The data below relate to the profits of Soko Yetu Groceries (in thousands of shillings) over a period of four years.

**Required:**

Determine the trend equation using the least squares method. (8 marks)

Calculate the seasonal index for each quarter using the multiplicative model. (6 marks)

**(Total: 20 marks)**

**QUESTION SEVEN**

1. Explain THREE decision making environments. (6 marks)

2. Define the following terms as used in decision making:

Value of perfect information. (1 mark)

Regret. . (1 mark)

3. Majux Limited manufactures two types of fruit juices; yellow juice and red juice. 1 packet of yellow juice requires 3 minutes for cutting of fruits, 6 minutes for blending, 7 minutes for cooling and 2 minutes for packaging. 1 packet of

red juice requires 5 minutes for cutting of fruits, 4 minutes for blending, 10 minutes for cooling and 5 minutes for packaging.

The company’s workforce can only spend a maximum of 60 hours on cutting, 711 /3 hours on blending, 105 hours on

cooling and 45 hours on packaging. The profit contribution is Sh.450 for each packet of yellow juice and Sh.380 for

each packet of red juice.

**Required:**

Formulate a linear programming model from the above information. (5 marks)

Use the graphical method to solve the linear programming model formulated in (c) (i) above. (5 marks)

Calculate the slack or surplus values for cutting of fruits and interpret its meaning. (2 marks)

**(Total: 20 marks)**