December 2024

1a

Mathematical Techniques - Calculus

Explain TWO applications of calculus in business management.

1b

Mathematical Techniques - Functions

Consider a company manufacturing a product with the following functions:

Marginal revenue (Sh.“000”) = 50 - 4q

Average costs (Sh.“000”) = 5 - 16/q

Required:

(i) Formulate the profit function.

(ii) Determine the quantity to be produced in order to maximise profit.

(iii) Calculate the maximum profit.

1c

Mathematical Techniques - Matrix Algebra

The 2 x 2 matrices; A, X and B are given as follows:

\(A = \begin{bmatrix} 10 & 4 \\ 4 & 2 \end {bmatrix}; X = \begin{bmatrix} a & b \\ c & d \end {bmatrix}; A = \begin{bmatrix} 18 & 14 \\ 8 & 10 \end {bmatrix};\)

Required: Given that AX = B, find the values of a, b, c and d.

2a

Probability

Outline FOUR characteristics of a poisson probability distribution.

2b

Probability

In relation to probability theory, distinguish between “subjective approach to probability” and “classical approach to probability”.

2c

Probability

A manufacturing company is evaluating the preferences of 250 customers regarding three new products; product Exe, product Wye and product Zed. The following information was gathered from a sample survey:

35 customers prefer both product Exe and product Wye.
120 customers prefer either product Exe or product Wye but not product Zed.
50 customers prefer product Wye but not product Zed or product Exe.
140 customers prefer either product Wye or product Zed but not product Exe.
70 customers prefer either product Zed but not product Exe or product Wye.
20 customers prefer both product Exe and product Zed but not product Wye.
45 customers prefer product Exe but not product Wye or product Zed.

Required:

(i) Visualise the customers’ preferences using a Venn diagram.

(ii) Determine the probability that a customer selected at random expressed a preference for all the three products.

(iii) Determine the probability that a customer selected at random expressed a preference for at most two of the three products.

(iv) Determine the probability that a customer selected at random did not prefer any of the three products.

3a

Time series

Explain THREE methods of estimating the trend line in time series analysis.

3b

Probability

The manager of Glaze Ltd. has selected a random sample of 150 invoices from the company’s outstanding invoices and provides the following distribution:

Sales Sh.“000”	Probability of outstanding invoice
0 – 100	0.20
100 – 200	0.18
200 – 300	0.22
300 – 400	0.16
400 – 500	0.09
500 – 600	0.08
600 – 700	0.04
700 – 800	0.03

Required:

(i) Calculate the expected mean of the random sample.

(ii) Calculate the expected standard deviation of the random sample.

(iii) Calculate the coefficient of variation of the random sample.

(iv) Determine the 95% confidence interval of the mean value of the outstanding invoices.

4a

Linear programming

Enumerate FOUR assumptions of linear programming model.

4b

Decision Theory

Faida Ltd. is in the process of reviewing the selling price for product “Excel”.

The selling prices under consideration are Sh.50, Sh.55 and Sh.60.

The following additional information is provided about the forecasted demand for product “Excel” under three different market conditions:

	Market condition (Demand in Units)
Selling price	Good	Moderate	Bad
Sh.50	20,000	18,000	14,000
Sh.55	18,000	16,500	12,000
Sh.60	16,000	14,000	8,500

The fixed costs are estimated at Sh.300,000 and variable cost per unit is Sh.30.

Required:

(i) Construct the pay-off trade for product “Excel”.

(ii) Advise Faida Ltd. on the best selling price for product “Excel” based on maximax criterion.

(iii) Advise Faida Ltd. on the best selling price for product “Excel” based on maximin criterion.

(iv) Determine the best selling price for product “Excel” based on expected monetary value (EMV) criterion given that probability for good, moderate and bad market criterion is 0.30, 0.50 and 0.20 respectively.

(v) Determine the expected value of perfect information.

5a

Mathematical Techniques - Functions

Describe FOUR applications of linear functions in business.

5b

Correlation and Regression Analysis

A research assistant wishes to determine how the weight of 10 students is related to their height. He collects a random sample of 10 students and measures both height in centimetres (cm) and their weight in kilograms (kg).

The results are shown in the table below:

Student Number	Height (Cm)	Weight (Kg.)
1	149	49.2
2	150	49.4
3	147	46.7
4	147	47.4
5	142	41.6
6	138	35.5
7	151	51.4
8	148	47.4
9	144	36.7
10	140	36.4

Required:

(i) The product moment correlation coefficient.

(ii) The ordinary least squares regression equation.

(iii) Predict the weight of a student with a height of 152cm.

(iv) The standard error of estimate.

(v) Determine at 95% confidence interval, the predicted weight of a student with a height of 152cm.

6a

Mathematical Techniques - Matrix Algebra

Using appropriate illustrations, explain the following terms as used in matrices:

(i) Diagonal matrix.

(ii) Scalar matrix.

6b

Linear programming

A factory produces two products; product A and product B. The company wants to determine how many units of each product to produce in order to maximise profit. The production of each product is subject to two resource constraints; machine hours and labour hours.

The profit for each unit of product A is Sh.40 and Sh.30 for product B.

Each unit of product A requires 2 hours of machine time, while each unit of product B requires 1 hour of machine time. The company has a maximum of 100 machine hours available.

Each unit of product A requires 1 hour of labour and each unit of product B requires 2 hours of labour. The company has a maximum of 80 labour hours available.

Required:

(i) Formulate the above problem as a linear programming problem.

(ii) Determine the optimal solution using the simplex method.

7a

Probability

Tangles Limited is the sole producer of three hair products; kinky, fluffy and flat that currently have a market

share of 30%, 50% and 20% respectively.

Each month, some brand switching takes place as follows:

Customers that bought kinky the previous month, 70% buy it again while equal proportion switch to fluffy and flat respectively.
Customers that bought fluffy the previous month, 60% buy it again while 25% switch to kinky and 15% to flat.
Customers that bought flat the previous month, 70% remain while 10% switch to kinky and 20% to fluffy.

Required:

(i) Construct a probability transition matrix of the switching probabilities.

(ii) Calculate the new market share a month after the current market share.

(iii) Calculate the steady state.

7b

Hypothesis Testing and Estimation

A company operates under two departments; production and marketing department. Details relating to a sample of employees working in the two departments has been provided as follows:

	Accounting Department	Marketing Department
Number of employees	40	50
Average monthly salary	Sh.120,000	Sh.130,000
Standard deviation	Sh.10,000	Sh.12,000

Required:

Determine whether there is any significant difference between the average salaries of employees working in the two departments. (Use a significance level of 5 percent).

December 2024

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