**WEDNESDAY: 26 April 2023. 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. A survey was conducted to establish the number of faulty components from a production process. The results indicated that 01% of the components produced were faulty. Each machine produces 10,000 components.

### Required:

The probability of there being 3 or more faulty components assuming a Poisson probability distribution. (4 marks)

2. The Sales Manager of XYZ Ltd. estimates that the revenue function of the company is linear in nature. He has observed that at a selling price of Sh.200, the company sold 6,000 units. When the company increased the selling price per unit to 700, the company sales decreased to 1,000 units.

The Cost Accountant of the company estimates that the cost function is quadratic in nature. The following data was provided:

**Required:**

- The revenue (4 marks)
- The total cost (8 marks)
- The profit maximising level of (4 marks)

### (Total: 20 marks)

** **

**QUESTION** **TWO**

1. A survey was recently conducted to determine the preferences of 360 customers with regard to three brands of cooking oil namely; sunflower oil, coconut oil and olive

The following results were obtained:

220 customers preferred sunflower oil. 160 customers preferred coconut oil. 180 customers preferred olive oil.

80 customers preferred both sunflower oil and coconut oil. 110 customers preferred both sunflower oil and olive oil. 100 customers preferred both coconut oil and olive oil.

50 customers preferred none of the brands of cooking oil.

### Required:

- Present the above information in the form of a Venn (4 marks)
- Determine the probability that a customer picked at random prefers all the three brands of cooking (2 marks)
- Determine the probability that a customer picked at random prefers at least two brands of cooking (2 marks)

2. The time taken to complete a particular task was measured for 250 workers and the results were as shown below:

### Required:

- Determine the value of X given that the median time taken to complete the task is 5 minutes. (6 marks)
- Determine the value of Y given that the modal time taken to complete the task is 5 minutes. (6 marks)

### (Total: 20 marks)

**QUESTION** **THREE**

1. Using the information given below, determine the level of output necessary from each sector to support a final demand of 500 units for sector X1, 1,600 units for sector X2 and 2,000 units for sector X3:

2. The monthly rent paid by residents of Hill View Estate is found to be normally distributed with a mean of 22,150 and a standard deviation of Sh.8,900.

### Required:

On the basis of a sample of 200 residents:

- Construct a 95% confidence interval for the monthly rent (3 marks)
- Test the hypothesis that the amount paid in monthly rent is Sh.20,000 at a 5% significance (5 marks)

### (Total: 20 marks)

**QUESTION FOUR**

1. Differentiate between the following terms:

- “Coefficient of skewness” and “coefficient of variation”. (4 marks)
- “Arithmetic mean” and “harmonic mean”. (4 marks)

2. Mr Rajab Omar is the Marketing Manager of Tintex He gathered data on the number of sales people engaged in a month and the sales made in the last ten months of the year 2022.

### Required:

- Estimate the total sales function using the ordinary least squares (10 marks)
- Predict the sales made in the month of January 2023 when 19 sales people were (2 marks)

### (Total: 20 marks)

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**QUESTION** **FIVE**

1. Outline **FOUR **properties of a (4 marks)

2. A trader purchased 7 cartons of apples and 4 cartons of bananas at a total cost of 32,500 in the month of January. In the month of February, he purchased 9 cartons of apples and 6 cartons of bananas at a total cost of Sh.43,500.

### Required:

Calculate the cost of a carton of apples and a carton of bananas using matrix algebra. (6 marks)

3. Red Rose Limited has developed a new washing The company is considering whether or not to market test the detergent prior to its launch. Market testing will cost Sh.50 million which may increase the likelihood of success of the product if it passes the test. The probability of passing the test is 0.8. If the product fails the test, it is regarded as worthless.

Red Rose Limited has the option of selling the product design for Sh.250 million.

### Required:

- A decision tree showing the pay-off and the expected monetary value (EMV) of each (8 marks)
- Advise the management of the company on the best course of action and the related expected monetary value (EMV). (2 marks)

### (Total: 20 marks)

**QUESTION SIX**

1. In the context of time series analysis, distinguish between the following terms:

- “Mean absolute deviation” and “mean squared error”. (4 marks)
- “Additive model” and “multiplicative model”. (4 marks)

2. The quarterly profits of Nyota Limited during the years 2019, 2020, and 2021 were as tabulated below:

### Required:

- The three-quarter moving (4 marks)
- The seasonal index for each quarter assuming the multiplicative (4 marks)
- Derive the deseasonalised profits using the multiplicative (4 marks)

### (Total: 20 marks)

** **

**QUESTION** **SEVEN**

1. Explain the meaning of the following terms in the context of probability theory:

- Mutually exclusive (2 marks)
- Independent (2 marks)

2. A linear programming problem has been formulated as below:

### Required:

- Optimal production for x and y using the simplex (12 marks)
- The slack values for each (2 marks)
- The shadow price for each (2 marks)