April 2025

1a

Decision Theory

Discuss THREE emerging issues in quantitative analysis that have significantly impacted decision making in modern industries.

1b

Mathematical Techniques

A company sells a product and market research shows that the demand function for the product is linear. The price and quantity combinations observed in the market are as follows:

When price (P) is Sh.50, the quantity demanded (Q) is 200 units.

When price (P) is Sh.30, the quantity demanded (Q) is 400 units.

The total cost function (C) is given by C(Q) = 5,000 + 10Q + 0.05Q^2

Required:

(i) Determine the linear demand function.

(ii) Find the total revenue function.

(iii) Find the profit function.

(iv) Determine the quantity that maximises profit.

(v) Determine the break-even point.

2a

Probability

Explain the following terms as used in probability theory:

(i) Conditional probability.

(ii) Bayes theorem.

2b

Mathematical Techniques - Descriptive Statistics

A company collected data on monthly sales (in units) of a particular product over the past 100 months. The data was grouped into eight classes as shown below but two frequency values (\(F_3\) and \(F_6\)) are missing. The mean sale is given as 48.3 units.

Sales range (units)	Frequency
10 – 20	8
20 – 30	12
30 – 40	\(F_3\)
40 – 50	20
50 – 60	18
60 – 70	\(F_6\)
70 – 80	10
80 – 90	7
Total	100

Required:

(i) Solve for \(F_3\) and \(F_6\) using the given mean.

(ii) Compute the median sales.

(iii) Determine the first quartile value (\(Q_1\)).

(iv) Determine the third quartile value (\(Q_3\)).

(v) Determine the quartile deviation.

(vi) Determine the quartile coefficient of skewness.

3

Time series

A company tracked its quarterly sales (in millions) over the past 3 years as shown in the table below:

	Quarter
Year	\(Q_1\)	\(Q_2\)	\(Q_3\)	\(Q_4\)
1	50	65	80	70
2	55	70	85	75
3	60	75	90	80

Required:

(a) Obtain a 4-quarter centred moving average for the data.

(b) Determine the typical seasonal indices for each quarter using the multiplicative model.

(c) Determine the deseasonalised sales for each quarter.

(d) Fit a trend equation to the deseasonalised data using the ordinary least squares (OLS) method.

(e) Forecast the seasonally adjusted sales for each quarter of the coming year using the trend equation in (d) above.

4a

Hypothesis Testing and Estimation

Outline the SIX steps involved in conducting a hypothesis test.

4b

Hypothesis Testing and Estimation

AutoTech Solutions Ltd. is analysing the relationship between total maintenance cost (Y) and two predictor variables: Service hours (X1) and number of replacement parts used (X2). Regression output from Ms Excel is as follows:

Regression statistics:
Parameter	Output
Multiple R	0.981542
R square	A?
Adjusted R square	0.955684
Standard error	24.137620
Observations	10

ANOVA Table:
Source	d.f	SS	MS	F	Significance
Regression	2	254,600.40	C = ?	D = ?	0.000089
Residual	7	B=?	3,456.21
Total	9	263,000

Regression coefficients:
Variable	Coefficient	Standard error	t-statics	P value	Lower 95%	Upper 95%
Intercept	480.32	30.1175	E = ?	0.000012	410.28	550.36
Service hours (\(X_1\))	F= ?	4.9162	5.847	0.000095	22.32	35.18
Replacement parts (\(X_2\))	15.60	3.2453	48.08	0.000231	10.32	20.88

Required:

(i) Determine the missing values A, B, C, D, E and F in the regression output.

(ii) Develop a regression equation showing the relationship between total maintenance cost and two predictor variables.

(iii) Determine the coefficient of determination. Interpret your results.

(iv) Test the adequacy of the model for prediction (F table value = 4.74).

(v) Explain whether service hours are adequate as predictor variable (t critical value = 2.365).

5a

Mathematical Techniques - Descriptive Statistics

Highlight SIX qualities of a good measure of central tendency.

5b

Decision Theory

Pegra Ltd. is considering launching a new electric product. However, demand for the proposed product is uncertain and the company can either launch the new product immediately or conduct market research before making a decision which could return either a favourable or unfavourable outcome. If the research outcome is favourable, Pegra Ltd. can proceed with launch. If the research outcome is unfavourable, the company has the option to abandon the launch.

Probabilities and payoffs:

Option 1:	Launch immediately: Probability of high demand will be 60% with a projected profit of Sh.500,000 Probability of low demand will be 40% with a projected loss of Sh.200,000
Option 2:	Conduct market research at a cost of Sh.50,000 Probability of favourable research outcome is 70% Probability of unfavourable research outcome is 30%

If research outcome is favourable:

Probability of high demand will be 80% with a projected profit of Sh.500,000
Probability of low demand will be 20% with a projected loss of Sh.200,000

If research outcome is unfavourable:

The company can choose to abandon the launch incurring only the Sh.50,000 research cost.

Required:

(i) Construct a decision tree based on the given probabilities and outcomes.

(ii) Compute the expected monetary value (EMV) for each option.

(iii) Recommend the best investment decision for the company.

6a

Probability

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

(i) Universal set.

(ii) Subset.

(iii) Set union.

6b

Mathematical Techniques

A simple economy consists of two sectors; Wool production and Hides tanning. Each unit of output from Wool production requires 0.6 units worth of input from Wool production and 0.3 units worth of input from Hides tanning.

A unit of output from Hides tanning requires 0.2 units worth of input from Wool production and 0.5 units worth of input from Hides tanning. The final demand for the period is estimated to be 800 units for Wool production and 220 units for Hides tanning.

Required:

(i) Determine the total output required for each sector to satisfy both intermediate and final demand.

(ii) Account for the usage of Wool production output.

(iii) Account for the sources of input for Hides tanning.

7a

Hypothesis Testing and Estimation

Explain the following terms as used in hypothesis testing:

(i) Type I error.

(ii) Type II error.

(iii) Rejection region.

7b

Linear programming

A company owns two flour mills; A and B which have different production capabilities for high, medium and low grade flour. The company has entered into a contract to supply flour to a firm every week with 120, 80 and 240 kilograms of high, medium and low grades flour respectively. It costs the company Sh.10,000 and Sh.8,000 per day to run Mill A and Mill B respectively. On a daily basis, Mill A produces 60, 20 and 40 kilograms of high, medium and low grade flour respectively while Mill B produces 20, 20 and 120 kilograms of high, medium and low grade flour respectively.

Required:

(i) Formulate the above problem as a linear programming problem in order to minimise the total cost of operation.

(ii) Graphically determine the number of days per week that each mill should operate to minimise the total cost of operation.

April 2025

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