QUANTITATIVE ANALYSIS APRIL 2022 PAST PAPER

WEDNESDAY: 6 April 2022. Morning paper.                                                                                          Time Allowed: 3 hours.

Answer any FIVE questions. ALL questions carry equal marks. Show ALL your workings. Do NOT write anything on this paper.

QUESTION ONE

1.            Hexadol Limited has been in operation for the last 5 years. The Company’s annual revenue function and annual cost function take the form of quadratic functions.

The following data was obtained from the records of the company for the last 3 years:

Required:

The revenue function of the company.                                                                                                            (4 marks)

The cost function of the company.                                                                                                                  (4 marks)

2.           Explain the following terms as used in probability:

Joint probability.                                                                                                                                               (2 marks)

Mutually exclusive events.                                                                                                                                (2 marks)

Conditional probability.                                                                                                                                   (2 marks)

Dependent events.                                                                                                                                             (2 marks)

3.           A firm has 500 employees out of whom, 2% have a minor accident in a given year. Out of the employees who have a minor accident in a given year,  30% had safety instructions. 80% of all employees had no safety instructions.

Required:

The probability of an employee being accident free given that the employee had no safety instructions.               (4 marks)

(Total: 20 marks)

QUESTION TWO

1.            Explain three types of Kurtosis that a graphical representation of a frequency distribution can assume.            (6 marks)

2.           The following data shows the age distribution of 350 employees of a multi-national company.

Age (years)          Number of employees

20 — 25                                    20

25 — 30                                    70

30 — 35                                  100

35 —40                                   65

40 — 45                                    40

45 — 50                                    25

50 — 55                                     15

55 — 60                                     10

60 — 65                                       5

Required:

Compute the following measures and give an interpretation of each:

The mean age.                                                                                                                                                 (2 marks)

The median age.                                                                                                                                             (3 marks)

The standard deviation of the employees’ ages.                                                                                    (6 marks)

The Karl Pearson’s coefficient of skewness.                                                                                          (3 marks)

(Total: 20 marks)

QUESTION THREE

1.          A random sample of 350 invoices issued by ARIK Ltd. revealed an average invoice value of Sh.38,780 with a standard deviation of Sh.8,750.

Required:

The interval within which the population mean should fall at 95% level of confidence.          (3 marks)

The sample size of invoices that would result in a 99% confidence level of the interval obtained in  above.                    (3 marks)

2.          The following information relates to the operating life of two brands of mobile phones, brand A and brand B:

Brand A                 Brand B

Mean life (days)                                        1,730                      1,684

Standard deviation (in days)                     102                         108

Sample size                                                    200                         200

Required:

Advise a potential customer on whether there is any significant difference between the quality of the two brands of mobile phones at a 5% level of significance.                                                                                                         (6 marks)

3.          A certain new strategy K was claimed to be effective for teams playing a certain game. In a marathon of 400 games, half of the teams used strategy K and the other half used strategy P.

The teams’ performance was recorded in the following table

Won                Defeated             Drawn

Adopted strategy K           130                     20                          60

Adopted strategy P           110                     30                          50

Required:

On the basis of the above data and using the chi-square method, advise whether there is a significant difference in the effect of the two strategies, K and P.                                                                                                                 (8 marks)

(Total: 20 marks)

QUESTION FOUR

1.            XYZ Ltd. produces three products namely A, B and C. The company presents the profit per unit of the products it produces and sells as follows:

Sh.2,000, Sh.3,000 and Sh.4,000 for products A, B and C respectively during the month of January 2022. Sh.7,000, Sh.9,000 and Sh.4,000 for products A, B and C respectively during the month of February 2022. Sh.1,000, Sh.4,000 and Sh.2,000 for products A, B and C respectively during the month of March 2022. The total profits in the months of January, February and March 2022 are Sh.17 million, Sh.45 million and Sh.16 million respectively.

Required:

The total number of each product produced and sold using matrix algebra.                                                (8 marks

2.           The savings accounts in a certain microfinance bank have an average balance of Sh.240,000 and a standard deviation of Sh.60,000. The account balances are assumed to be normally distributed.

Required:

The proportion of savings accounts whose balances are above Sh.275,000.                                (3 marks)

The proportion of savings accounts whose balances lies between Sh.190,000 and Sh.260,000.                                (3 marks)

The average revenue function of a certain company is given by the function AR = 2,000 — 24q. The cost function is given by the function C = 6q2 + 1,440q + 1,280. In both cases, q represents the quantity in units.

Required:

The profit function of the company.                                                                                                        (3 marks)

The maximum profit for the company.                                                                                                   (3 marks)

(Total: 20 marks)

QUESTION FIVE

Ahadi Ltd. is in the process of analysing its electricity expense and its relationship with the machine hours of operation. The following data is provided with respect to the year ended 31 December 2021:

Month              Number of machine hours             Electricity expense

“000”                                       Sh.”000″

January                                    72                                              1,020

February                                  55                                                 820

March                                      39                                                 720

April                                         60                                                 900

May                                          49                                                  870

June                                          39                                                 720

July                                           53                                                 825

August                                     81                                              1,365

September                              63                                                 870

October                                    59                                                 890

November                               45                                                  790

December                               50                                                 940

Required:

The least squares regression line for the above data and interpret its meaning.                                       (10 marks)

Estimate the amount of electricity expense assuming the expected machine hours are 78,000.            (2 marks)

The product moment correlation coefficient between machine hours and electricity expense. Interpret your answer. (6 marks)

4.          The standard error of estimate for the regression line. Interpret your answer.                                           (2 marks)

(Total: 20 marks)

QUESTION SIX

1.          Highlight four requirements that must be met before the linear programming model can be applied. (4 marks)

A company makes two products; 1 and 2.

Each product requires time on two machines A and B. The specifications for each product are as follows:

Product 1       Product 2

Processing time:

Machine A (Hours/unit)                                         1.6                  1.0

Machine B (Hours/unit)                                        2.5                   1.0

Selling price (Sh./unit)                                           22                   48

Material and labour cost (Sh./unit)                      14                   37

Maximum possible production and sale (units)  30                   50

Maximum time available for machine A is 80 hours and for machine B is 100 hours.

Required:

Formulate a linear programming model to determine the number of product    1 and product 2 which should be produced and sold in order to maximise total contribution for the company using the graphical method. (12 marks)

2.          State any four assumptions of the Poisson probability distribution.                                                                                    (4 marks)

(Total: 20 marks)

QUESTION SEVEN

1.            The table below shows the quarterly profits of Kahawa Limited (in millions of shillings) for the years 2019, 2020 and 2021:

Quarterly profits (Sh.”million”)

Year  Quarter 1   Quarter 2       Quarter 3      Quarter 4

2019               23                    32                    27                    21

2020               27                    35                    32                    24

2021               31                    43                    40                    29

Required:

The three-quarter moving average of the profits.                                                                                                (6 marks)

The quarterly seasonal variations of the profits using the additive model.                                                           (4 marks)

Forecast the adjusted profits for the year 2022 given that the actual profits (in Sh.”million”) in the year 2022 are 35, 50, 47 and 33 for Quarter 1, Quarter 2, Quarter 3 and Quarter 4 respectively.                               (4 marks)

2.      An investment manager in an investment fund has a choice between:

1. A diversified portfolio promising Sh.15 million with a probability of 0.7 and Sh.8 million with a probability of 0.3.
2. A risky investment consisting of two contracts with independent outcomes one promising Sh.7 million with a probability of 0.7 and the other Sh.3.5 million with a probability of 0.3.

Required:

Construct a decision tree depicting the above information using the expected monetary value (EMV) criterion.                         (3 marks)

Advise on the best decision using the EMV criterion.                                                                                        (3 marks)

(Total: 20 marks)

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