### WEDNESDAY: 23 August 2023. 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. Explain the meaning of the following terms as used in decision theory:

Decision alternative. (2 marks)

State of nature. (2 marks)

Conditional payoff. (2 marks)

Opportunity cost. (2 marks)

2. The following data relates to the ages of 100 students attending a workshop on personal branding organised by the student welfare officials of Pride Business College:

Thereafter, 15% of the youngest students and 5% of the oldest students attending the workshop were selected to attend a further training on curriculum vitae writing.

**Required:**

Determine the youngest age of the students selected to attend the training on curriculum vitae writing.

(4 marks)

Determine the highest age of the students selected to attend the training on curriculum vitae writing.

(4 marks)

Calculate the median age of the students who remained after the selection of students to attend the

training on curriculum vitae writing. (4 marks)

**(Total: 20 marks)**

**QUESTION TWO**

1. State FOUR applications of mathematical functions in business. (4 marks)

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

Disjoint set. (2 marks)

Complement of a set. (2 marks)

Union of a set. (2 marks)

3. The following regression equation was obtained for a class of 24 intermediate level students:

**Required:**

Calculate the t ratio and the 95% confidence interval for the independent variables X1, X2 and X3.

(6 marks)

Determine the regressor which gives the strongest evidence of being statistically discernible. (2 marks)

In writing up a final regression, explain whether one should keep the last regressor (X3) in the equation or drop it. (2 marks)

**(Total: 20 marks)**

**QUESTION THREE**

1. State FOUR characteristics of the normal distribution. (4 marks)

2. A firm manufactures two models of bicycles; mountain bike and BMX. The firm earns profit of Sh.5,000 and Sh.6,000 on mountain bikes and BMX respectively. Both models are produced in three departments; assembly, fitting and painting. The time required per model produced and the time available per week (in hours) are given in the table below:

**Required:**

Formulate the above problem as a linear programming problem in order to maximise profits. (4 marks)

Graphically show how the manufacturer should schedule his production to maximise profits. (8 marks)

Compute and interpret the slack value for the painting department. (4 marks)

**(Total: 20 marks)**

**QUESTION FOUR**

1. Explain the following terms as used in Markovian analysis:

Transition matrix. (2 marks)

Equilibrium state. (2 marks)

Initial probability vector. (2 marks)

2. The following pay-off matrix was developed by a company showing profits (in shillings) obtained from launching four different products P1, P2, P3 and P4 under four different states of nature:

The probabilities for S1, S2, S3 and S4 are given as 0.30, 0.40, 0.20 and 0.10 respectively.

**Required:**

Advise on the best course of action using the Mini-Max Regret Criterion. (4 marks)

Advise on the best course of action using the Expected Opportunity Loss Criterion. (4 marks)

An expert has offered to provide perfect information at a cost of Sh.2,500.

Advise the management of the company on whether or not to acquire the perfect information. (6 marks)

**(Total: 20 marks)**

**QUESTION FIVE**

1. The output of an acre of land is assumed to be normally distributed with an average of 52 bags of maize and a standard deviation of 3.2 bags.

**Required:**

The probability that the output of an acre of land:

Is greater than 48 bags. (2 marks)

Is greater than 60 bags. (2 marks)

Is less than 45 bags. (2 marks)

Lies between 50 bags and 60 bags. (2 marks)

2. BMM Limited produces X number of items of product “Wonder” in a month at a cost described by the equation C = 5x + 4,000. The Management Accountant of the firm estimates that at a selling price of Sh.22 per unit, 18,000 units of “Wonder” could be sold. If the firm increases the unit price to Sh.30, only 10,000 units of “Wonder” can be sold.

**Required:**

Determine the number of units of product “Wonder” that BMM Limited should produce and sell in order

to maximise profit. (6 marks)

Determine the selling price per unit charged at the maximum profit. (2 marks)

Calculate the break-even number of units. (4 marks)

**(Total: 20 marks)**

**QUESTION SIX**

1. Distinguish between a “two-tailed test” and a “one tailed test” as used in inferential statistics. (4 marks)

2. The data below shows the sales made by Kuza Limited over a period of 6 years:

**Required:**

The sales forecast for the year 2023 using exponential smoothing (use a smoothing constant of 0.2).

(4 marks)

The sales forecast for the year 2023 using the ordinary least squares method. (6 marks)

Using suitable computations, advise Kuza Ltd. on the preferred forecast method. (6 marks)

**(Total: 20 marks)**

**QUESTION SEVEN**

1. With the aid of diagrams, describe the THREE types of Kurtosis. (6 marks)

2. Consider the following hypothesis:

For a random sample of 12 observations, the sample mean was 407 and the sample standard deviation was 6.

**Required:**

Using a significance level of 0.1, advise whether the null hypothesis should be accepted or rejected. (6 marks)

3. A mobile phone manufacturer orders for a special component called PH-2 from four different suppliers; S1, S2, S3 and S4. 20% of the components are purchased from S1, 10% from S2, 30% from S3 and the remainder from S4.

From past experience, the manufacturer knows that 2% of the components from S1 are defective, 4% of the components from S2 are defective, 3% of the components from S3 are defective and 1% of the components from S4 are defective. All components are placed directly in the store before inspection. A worker selects a component for use and finds it defective.

**Required:**

The probability that the component was supplied by S1. (4 marks)

The probability that the component was supplied by S2 or S4. (4 marks)

**(Total: 20 marks)**