August 2023

1a

Decision Theory

Explain the meaning of the following terms as used in decision theory:

(i) Decision alternative.

(ii) State of nature.

(iii) Conditional payoff.

(iv) Opportunity cost

1b

Mathematical Techniques - Descriptive Statistics

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:

Age (in years)	Number of students
Below 20	2
20 – 25	4
25 – 30	10
30 – 35	20
35 – 40	32
40 – 45	18
45 – 50	10
Above 50	4

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:

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

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

(iii) Calculate the median age of the students who remained after the selection of students to attend the training on curriculum vitae writing.

2a

Mathematical Techniques

State FOUR applications of mathematical functions in business.

2b

Decision Theory

Explain the following terms as used in set theory:

(i) Disjoint set.

(ii) Complement of a set.

(iii) Union of a set

2c

Correlation and Regression Analysis

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

\(\hat{Y}= 4.3 + 0.029X_1 + 0.029X_2 + 0.017X_3\)

Standard error

0.0074

0.013

0.007

Where:

ŷ = Students score on a theory examination

\(X_1\) = Students rank (from the bottom) in high school

\(X_2\) = Students verbal aptitude score

\(X_3\) = A measure of student character

Required:

(i) Calculate the t ratio and the 95% confidence interval for the independent variables \(X_1, X_2 \text{ and } X_3\)

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

(iii) In writing up a final regression, explain whether one should keep the last regressor (\(X_3\)) in the equation or drop it.

3a

Decision Theory

State FOUR characteristics of the normal distribution.

3b

Linear programming

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:

Departments	Required time		Available time
	Mountain bike	BMX
Assembly	2	3	180
Fitting	2	1	120
Painting	3	3	240

Required:

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

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

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

4a

Probability

Explain the following terms as used in Markovian analysis:

(i) Transition matrix.

(ii) Equilibrium state.

(iii) Initial probability vector.

4b

Decision Theory

The following pay-off matrix was developed by a company showing profits (in shillings) obtained from launching four different products P_1, P_2, P_3 and P_4 under four different states of nature:

State of nature
Product	\(S_1\)	\(S_2\)	\(S_3\)	\(S_4\)
\(P_1\)	5,000	9,000	7,000	8,000
\(P_2\)	7,000	4,000	6,000	12,000
\(P_3\)	10,000	8,000	9,000	7,000
\(P_4\)	14,000	5,000	7,000	6,000

The probabilities for \(S_1\), \(S_2\), \(S_3\) and \(S_4\) are given as 0.30, 0.40, 0.20 and 0.10 respectively.

Required:

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

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

(iii) 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.

5a

Probability

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:

(i) Is greater than 48 bags.

(ii) Is greater than 60 bags.

(iii) Is less than 45 bags.

(iv) Lies between 50 bags and 60 bags.

5b

Mathematical Techniques

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:

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

to maximise profit.

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

(iii) Calculate the break-even number of units.

6a

Hypothesis Testing and Estimation

Distinguish between a “two-tailed test” and a “one tailed test” as used in inferential statistics.

6b

Time series

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

Year	2017	2018	2019	2020	2021	2022
Sales (in millions of shillings)	80	78	72	68	70	82

Required:

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

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

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

7a

Mathematical Techniques - Descriptive Statistics

With the aid of diagrams, describe the THREE types of Kurtosis.

7b

Hypothesis Testing and Estimation

Consider the following hypothesis:

\(H_0: \mu = 400\)

\(H_1: \mu \neq 400\)

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

7c

Probability

A mobile phone manufacturer orders for a special component called PH-2 from four different suppliers; \(S_1, S_2, S_3\) and \(S_4\). 20% of the components are purchased from \(S_1\), 10% from \(S_2\), 30% from \(S_3\) and the remainder from \(S_4\).

From past experience, the manufacturer knows that 2% of the components from \(S_1\) are defective, 4% of the

components from \(S_2\) are defective, 3% of the components from \(S_3\) are defective and 1% of the components from \(S_4\) are defective. All components are placed directly in the store before inspection. A worker selects a component for use and finds it defective.

Required:

(i) The probability that the component was supplied by \(S_1\).

(ii) The probability that the component was supplied by \(S_2\) or \(S_4\)

August 2023

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