**FRIDAY: 17 December 2021.** **Time Allowed: 3 hours.**

**Answer any FIVE questions. ALL questions carry equal marks. Show ALL your workings.**

**QUESTION ONE**

1. Explain the following terms as used in time series analysis:

Cyclical variations. (2 marks)

Random variations. (2 marks)

Seasonal variations. (2 marks)

Trend. (2 marks)

2. The following data relates to the profits reported by XYZ Ltd. in each of the months in the year 2020:

**Month** **Profit (Sh.”million”)**

January 40

February 38

March 39

April 41

May 36

June 41

July 34

August 37

September 35

October 37

November 40

December 41

**Required:**

Estimate the trend line using the ordinary least squares method. (9 marks)

Estimate the profit reported in March of the year 2021. (3 marks)

**(Total:** **20 marks)**

**QUESTION TWO**

1. State five advantages of the arithmetic mean as a measure of central tendency. (5 marks)

2. The following data shows the distribution of profits of 150 manufacturing companies in a given year:

**Profit (“Sh.”million”)** **Number of companies**

10-20 15

20-30 13

30-40 25

40-50 30

50-60 16

60-70 10

70-80 22

80-90 12

90-100 7

**Required:**

The arithmetic mean of the profits and interpret the results. (4 marks)

The standard deviation of the profits and interpret the results. (8 marks)

The coefficient of variation of the profits and interpret the results. (3 marks)

**(Total:** **20 marks)**

**QUESTION THREE**

1. Explain the following terms as used in Markov analysis:

Markov process. (2 marks)

Equilibrium state. (2 marks)

Absorbing state. (2 marks)

Closed state. (2 marks)

2. The marketing department of Jacuzi Ltd. estimates the daily demand function of one of its products to be linear in nature. If the price was fixed at Sh.570, the daily demand would be 400 units. If the price was increased to Sh.820, the daily sales would drop to 200 units.

The production department has indicated that the marginal cost of producing Q units of the product is given by the

following equation: MC = 2Q – 570

Where: MC is the marginal cost and

Q is the number of units produced.

The daily fixed cost is Sh.1,100.

**Required:**

The revenue function of Jacuzi Ltd. (4 marks)

The total cost function of Jacuzi Ltd. (3 marks)

The maximum profit that Jacuzi Ltd. could make. (5 marks)

**(Total: 20 marks)**

**QUESTION FOUR**

1. In the context of hypothesis testing, distinguish between a “type I error” and a “type II error”. (4 marks)

2. The sales before and after a promotional campaign in ten different regions for a certain commodity were recorded as follows:

**Region** **Sales before promotional** **Sales after promotional**

** campaign “Sh.million”** **campaign “Sh.million”**

1 53 58

2 28 29

3 31 30

4 48 50

5 50 50

6 42 45

7 63 59

8 40 36

9 25 22

10 30 28

**Required:**

Using a 5% level of significance, determine whether the promotional campaign was a success or not. (16 marks)

**(Total: 20 marks)**

**QUESTION FIVE**

Bantu Limited makes two types of pudding: vanilla and chocolate. Each serving of vanilla pudding requires 2 teaspoons of sugar and 25 fluid measures of water, and each serving of chocolate pudding requires 3 teaspoons of sugar and 15 fluid measures of water. Bantu Limited has available each day 3,600 teaspoons of sugar and 22,500 fluid measures of water. Bantu Limited makes no more than 600 servings of vanilla pudding because that is all that it can sell each day. Bantu Limited makes a profit of Sh.10 on each serving of vanilla pudding and Sh.7 on each serving of chocolate pudding.

**Required:**

1. Formulate a linear programming model to solve the above problem. (4 marks)

2. Construct an initial simplex tableau. (4 marks)

3. Using the simplex method, determine how many servings of each type of pudding Bantu Limited should make in order to maximise profit. (12 marks)

**(Total: 20 marks)**

**QUESTION SIX**

1. State four applications of matrices in business. (4 marks)

2. A global conference on “the blue economy” was recently held in Kenya and was attended by 280 delegates from America, Europe and Africa.

The following information relates to the delegates who attended the conference:

70 delegates represented Europe

96 delegates represented Africa

128 delegates represented America

20 delegates represented all the three continents.

25 delegates represented America and Africa

22 delegates represented America and Europe

26 delegates represented Europe and Africa

**Required:**

Present the above information in the form of a Venn diagram. (4 marks)

The number of delegates who represented at least two continents. (2 marks)

The number of delegates who represented only one continent. (2 marks)

The number of delegates who represented none of the three continents. (2 marks)

3. During the manufacture of a product, 0.002 of the product turns out to be defective. The product is supplied in packets of 10. A consignment of 100,000 packets is produced in a certain period.

**Required:**

Using the Poisson distribution, calculate the approximate number of packets containing:

No defectives. (2 marks)

1 defective. (2 marks)

2 defectives. (2 marks)

**(Total: 20 marks)**

**QUESTION SEVEN**

1. A random sample of 15 employees of a call centre was taken and each employee took a competency test. The mean of the scores achieved by these employees was 56.3% with a standard deviation of 7.1%. The results of this test have been found to be normally distributed in the past.

**Required:**

Construct a 95% confidence interval for the mean of the test score of the call centre employees. (6 marks)

2. Distinguish between the “coefficient of correlation” and the “coefficient of determination”. (4 marks)

The following data was obtained during a social survey conducted in a given urban area regarding the monthly income of households and their corresponding expenditure:

**Household** **Monthly** **Monthly**

** income** **expenditure**

** Sh.”000″** **Sh.”000″**

A 150 120

B 130 135

C 200 195

D 245 190

E 140 120

F 100 85

G 80 65

H 145 130

I 130 60

J 90 75

**Required:**

The Pearson’s coefficient of correlation between monthly income and monthly expeqditure and interpret the result. (10 marks)

**(Total: 20 marks)**