November 2020
Unit: Quantitative Analysis
13 Questions
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Questions
1a
Mathematical Techniques - Functions
A potential investor in the production of a new type of organic fertilizer estimated the demand function of the product to be AR = 150 - Q
Where:
AR is the average revenue in thousands of shillings.
Q is the output in tonnes.
The investor estimated the variable cost (VC) per unit tonne associated with the production to be.
VC/tonne = Q - 285 in thousands of shillings.
The firm's cost when not producing any output is estimated at Sh. 8,750,000.
Required:
(i). The profit furiction.
(ii). The level of output that maximises profit.
(iii). The breakeven output
1b
Decision Theory
A game between two players, A and B lias the following pay off matrix:
Required:
Required:
(i). The optimum strategy for each player.
(ii). The saddle point
(iii). The value of the game.
1c
Hypothesis Testing and Estimation
An accounting college has two classes, day class and evening class. From a survey conducted by the head of academics in the college, the following results were obtained:
| Classes | ||
| Day | Evening | |
| Number of students | 13 | 15 |
| Average test mark (%) | 45 | 55 |
| Standard deviation (%) | 4 | 5 |
Required:
Determine whether there is any significant difference in the average test mark between the two classes at 5% level of significance
2a
Decision Theory
A baker must decide whether to bake brown bread or white bread for a new market. Demand at the market can either be small or large with probability estimated to be 0.3 and 0.7 for brown bread and white bread respectively.
Additional information:
1. If brown bread is baked and demand proves to be high, the baker may choose not to expand (pay off = Sh.350,000) or to expand (pay off = Sh.420,000).
2. If brown bread is baked and demand is low, there is no reason to expand and the payoff is Sh.310,000.
3. If white bread is baked and demand proves to be low, the choice is to do nothing (Sh.90,000) or to stimulate demand through local advertising. The response to advertising may be either modest or sizeable, with their probabilities estimated to be 0.4 and 0.6 respectively. If it is modest, the pay off is estimated to be Sh.50.000 the pay off grows to Sh.340,000 if the response is sizeable.
4. If white bread is baked and the demand turns out to be high, the payoff is Sh. 1,400,000.
Required:
(i). A decision tree showing the payoff and expected monetary value of cach alternative decision.
(ii). Advise the management of the bakery on the best product to introduce into the market
2b
Correlation Analysis
In a choral music competition, 9 contestants were awarded marks in percentage using a music scoringuid by two assessors. The results obtained were given as shown in the table below:
Required:
(i). The rank correlation coefficient. Interpret your results.
(ii). Coefficient of determination
| Marks in % by: | ||
| Contestant | 1ˢᵗ Assessor | 2ⁿᵈ Assessor |
| A | 72 | 76 |
| B | 82 | 80 |
| C | 79 | 78 |
| D | 70 | 73 |
| E | 67 | 70 |
| F | 81 | 85 |
| G | 78 | 69 |
| H | 75 | 83 |
| I | 65 | 68 |
Required:
(i). The rank correlation coefficient. Interpret your results.
(ii). Coefficient of determination
2c
Probability
In a certain hospital, the arrival rate of patients into the outpatient department is 3 patients per hour and 4 patients are normally attended per hour.
Required:
(i). Service rate.
(ii). Length of queue.
(iii). Length of the system.
(iv). The time a patient takes being actually attended.
(v). The probability that there are more than six patients in the outpatient hospital department.
3a
Decision Theory
Dolce Ltd. is in the process of launching a new product into the market. Three variables are uncertain, selling price, variable cost and sales volume.
The following information is provided
| Selling price (Sh) 600 700 800 Variable cost (Sh.) 300 400 500 Sales volume (units) 40,000 50,000 60,000 | Probability 0.30 0.50 0.20 Probability 0.40 0.50 0.10 Probability 0.30 0.50 0.20 |
The following random numbers have been provided:
44, 84, 82, 50, 85, 40, 96, 88, 16, 16, 97, 92, 44, 82, 39, 33, 83, 42, 16, 07, 77 , 66, 50, 20, 50, 95, 83, 39, 58, 44, 77, 11, 08, 38, 89, 45, 09, 99, 81, 97, 50, 83,
Required:
The average contribution of Dolce Ltd using Monte Carlo simulation with 10 simulations.
3b
Regression Analysis
The production manager of Sweet Ltd. is concerned with the fluctuating indirect labour cost in relation to the labour hours worked by the employees.
The following data was collected for the past 12 months.
| Month | Labour hours "000" | Indirect labour cost "000" |
| January February March April May June July August September October November December | 48 68 94 82 46 78 96 60 72 62 88 68 | 963 752 1,032 1,316 710 1,180 1,456 770 1,004 1,211 917 1,190 |
Required:
Using the ordinary least squares method:
(i). Formulate the indirect cost function.
(ii). Compute the indirect labour cost for 120 labour hours.
(iii). Calculate the coefficient of determination.
4a
Linear programming
Explain the following terms as used in linear programming:
(i). Infeasibility.
(ii). Unboundedness.
4b
Linear programming
A training institution has four lecturers represented as L1, L2, L3 and L4. The Head of department wishes to assign them to handle three topics in quantitative analysis; T1, T2 and T3. This will be done based on competency which is measured in terms of mastery of subject matter and personal preference on the time schedule while satisfying policies and provisions of the institution.
All of the lecturers have taught the topics in the past and have been evaluated with the following scores in the three different topics as follows:
| Topics | ||||
| T1 | T2 | T3 | ||
| Lecturers | L1 L2 L3 L4 | 42 48 50 58 | 16 40 18 38 | 27 25 36 60 |
Required:
(i). The optimal assignment for these three topics,
(ii). The maximum score.
(iii). The lecturer that will not be assigned any topic.
4c
Time series
The data given below shows the profits in shillings million made by an economic sector in your country during the various quarters of the given years.
| Profits in quarters | ||||
| Year 2016 2017 2018 2019 | Q1 83 105 140 168 | Q2 260 383 430 503 | Q3 215 248 323 340 | Q4 393 553 588 755 |
Required:
(i). 3 quarter moving average of the series.
(ii). The deseasonalised profit of the economic sector using the additive model.
5a
Hypothesis Testing and Estimation
With reference to analysis of variances (ANOVA) tests:
(i). Distinguish between one-way and two-way ANOVA tests.
(ii). Outline four assumptions of two-way ANOVA tests.
(iii). Explain the difference between ANOVA tests and T-tests.
5b
Decision Theory
In the context of critical path analysis (CPA) method:
(i). Discuss two strengths and two weaknesses of CPA method.
(ii). Explain three practical applications of CPA method.
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