KASNEB Quantitative Analysis - Question 1a

State FIVE characteristics of binomial distribution ... [Click to view full question and step-by-step solution]

KASNEB Quantitative Analysis - Question 1c

Given the following sets: A = {a, b, c, d} B = {c, d, e, f, g} C = {h, i, j, c, d} Required: Find: (i) The universal set “⋃”. (ii) A⋂B⋂C. (iii) C' . ... [Click to view full question and step-by-step solution]

KASNEB Quantitative Analysis - Question 4a

Differentiate between the “multiplicative model” and “additive model” as used in time series ... [Click to view full question and step-by-step solution]

April 2024

1a

Probability

State FIVE characteristics of binomial distribution

1b

Correlation and Regression Analysis

Isabella Koech, a businesswoman, is contemplating investing in one of two manufacturing companies namely; company A and company B. The data below shows the profits made by company A and company B over the past 150 days.

Daily profits	Number of days
Sh.“000”	Company A	Company B
50 – 60	12	11
60 – 70	26	14
70 – 80	38	20
80 – 90	36	23
90 – 100	18	45
100 – 110	11	23
110 – 120	9	14

Required:

(i) Calculate the standard deviation for company A and company B.

(ii) Calculate the coefficient of variation for company A and company B.

(iii) Advise Isabella Koech on the company to invest in based on the results obtained in (b) (i) and (b) (ii) above.

1c

Probability

Given the following sets:

A = {a, b, c, d}

B = {c, d, e, f, g}

C = {h, i, j, c, d}

Required:

Find:

(i) The universal set “⋃”.

(ii) A⋂B⋂C.

(iii) C' .

2a

Mathematical Techniques

TOC Limited, an oil processing company purchased a machine that they intend to use in one of their production processes. The machine is expected to have a useful life of 10 years after which it can be disposed for Sh.200,000.

The net book value of the machine after 8 years is estimated to be Sh.560,000.

Required:

(i) The linear depreciation equation for the value of the machine.

(ii) Determine the value of the machine after 6 years.

(iii) Determine the number of years when the value of the machine will be Sh.1,460,000.

2b

Hypothesis Testing and Estimation

A production manager is convinced that a new set of plant engineers will not affect the production level in the factory. To test this hypothesis, 15 samples of the first day output is taken and average production per day is measured as 14,500 units with a standard deviation of 102. Before the new plant engineers were engaged, the average output per day was 14,650 units.

Required:

Test the manager’s conviction at 5% level of significance

2c

Decision Theory

Highlight FOUR considerations in determining whether to employ a census or a sample in collecting primary data.

3a

Mathematical Techniques

Clekex Cleaning Limited manufactures bar soaps for a hotel industry. The company specialises in three different types of bar soaps weighing 0.5kg, 0.75kg and 1kg respectively. The three types of bar soaps are made by two machines, machine A and machine B. The number of hours required for each type of bar soap in each machine is shown below:

Type of bar soap	Time taken for bar soap production (hours)
	Machine A	Machine B
0.5	0.40	0.25
0.75	0.60	0.12
1	0.50	0.60

Machine A has 2,800 hours available while machine B has 1,090 hours available. The management has decided that 1,000 bar soaps of 0.5kg must be produced.

Required:

The number of 0.75kg bar soaps and 1kg bar soaps that will be produced assuming machine A and B hours are fully utilised.

3b

Correlation and Regression Analysis

The following table shows the total cost per day on production of iron sheets for the last 10 days in a manufacturing company:

Day	Number of iron sheets produced	Total cost Sh.“000”
1	31	35
2	29	33
3	35	35
4	29	36
5	38	40
6	28	34
7	32	35
8	34	37
9	26	32
10	25	25

Required:

(i) Determine the total cost function using the ordinary least squares method.

(ii) Determine the break-even number of iron sheets produced assuming that each iron sheet is sold at Sh.2,200 each.

(iii) The coefficient of determination of the data.

(iv) Interpret the result in (c) (iii) above.

4a

Time series

Differentiate between the “multiplicative model” and “additive model” as used in time series

4b

Probability

A research company has established that the occurrence of a contagious disease follows a Poisson distribution with a mean of 0.3 per week.

Required:

(i) The probability that no case of a contagious disease is reported.

(ii) The probability that almost one case of a contagious disease is reported.

4c

Mathematical Techniques - Differentiation

A solar panel production firm intends to research on a launch of a new solar panel which would increase the sales of the panels to between 2,000 units to 3,000 units per week. The weekly revenue in thousands of shillings over the range of sales could be represented by:

R=−3x2+7x

Where x is the weekly solar panel units produced and sold in thousands. Past records of the solar panel production in the firm estimates its marginal costs in thousands of shillings could be represented by the function;

MC=2x2−3x+5

The fixed costs will be Sh.1,000 per week.

Required:

(i) The average cost function of the firm.

(ii) The average revenue function of the firm.

(iii) The profit maximising output.

(iv) The price that should be charged to maximise profits

5a

Time series

A car hire company has provided the data below showing the number of cars hired out for the last 10 months ending 31 March 2024:

Month	Number of cars
June 2023	110
July 2023	115
August 2023	109
September 2023	108
October 2023	106
November 2023	111
December 2023	107
January 2024	112
February 2024	114
March 2024	110

Required:

(i) The forecast number of cars for the month of April 2024 using exponential smoothing method with a smoothing constant α = 0.3.

(ii) The three months moving average forecasts for the month of April 2024.

(iii) Using suitable computation, advise on the best forecast method.

5b

Linear programming

World K Tours has Sh.6 million that may be used to purchase new rental minibuses for hire during the coming holidays. The minibuses may be purchased from two different manufacturers.

Important data concerning the minibuses is summarised below:

Minibus type	Manufacturer	Cost	Maximum seating	Expected daily profit
			capacity	per minibus (Sh.)
Weaverbird	Fastbus	80,000	11	10,000
Eaglet	Fastbus	90,000	14	12,000
Dovey	Smartbus	70,000	7	9,000
Crowlet	Smartbus	140,000	18	16,000

World K Tours wishes to purchase at least 80 minibuses and equal numbers from each of the manufacturer. World K Tours wishes to have a total sitting capacity of at least 500.

Required:

(i) Formulate the above linear programming problem.

(ii) Outline FOUR assumptions of the linear programming technique.

6a

Probability

A manager must select 4 employees for a job promotion. 12 employees are eligible for job promotion.

Required:

(i) Determine the number of ways in which 4 employees could be chosen.

(ii) Determine the number of ways in which the 4 employees could be chosen from the 12 employees in the department.

6b

Decision Theory

A property developer has Sh.1,000,000 to invest in a new property. The developer has a choice of three different properties going at the same price in different towns; one in Nairobi, another in Nakuru and the other in Nanyuki.

With the three properties, his ultimate profit depends on whether the economy is strong, weak or mixed.

He estimates that the pay-off matrix representing his monthly profits in thousands of shillings is as follows:

		States of nature
		Weak	Mixed	Strong
	Nairobi	85	30	75
Strategies	Nakuru	45	45	110
	Nanyuki	60	95	85

Required:

Determine the town that the developer should invest in using:

(i) Maxi-max criterion.

(ii) Maxi-min criterion.

(iii) Mini-max regret criterion.

(iv) Suppose the investor reads in the newspaper that there is 50% chance of a weak economy, 20% chance of a mixed economy and a 30% chance of a strong economy. Advise on the best strategy, given the new information.

(v) Compute the expected value of perfect information.

7a

Probability

A training institution requires that all its employees participate in one of three sports clubs namely; football, rugby or chess. In each term, the training institution designates an open selection period within which employees may change from one club to another.

Prior to the last open selection period, 10% of the employees preferred football, 35% preferred rugby and the remaining chess. During the open selection period, 25% of the employees taking football switched to rugby, while 15% switched to chess. 30% of the employees taking rugby switched to football and 10% switched to chess, 20% of the employees taking chess switched to football and 10% switched to rugby.

Required:

(i) The transition matrix.

(ii) The percentage of employees that will be taking each sport after the last open selection period.

(iii) Assuming that the trend continues, determine the percentage of the employees who will be taking each sport in the long-run.

7b

Probability

Agritech Consultants have engaged 3 farming trainees; Jim, Ken and Lorn. After six months, the Consultancy

supervisor noted that 5%, 7% and 9% of land planted by Jim, Ken and Lorn respectively had ungerminated plants.

The supervisor had distributed 40%, 25% and 35% of the land respectively.

Required:

(i) Present the above information in a tree diagram.

(ii) The probability of land with ungerminated plants.

(iii) The probability that land with ungerminated plants was planted by either Ken or Lorn

April 2024

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