MBA60: ADVANCED QUANTITATIVE METHODS FOR MANAGERS

Assignment 2

2008/9

HELLENIC OPEN UNIVERSITY: MBA PROGRAM

MBA60: ADVANCED QUANTITATIVE METHODS FOR MANAGERS

Advanced Quantitative Analysis - 2nd Assignment

Exercise 1:

Α gasoline additive is being tested to see whether it increases mileage. Twenty-five cars are supplied with 5 litres of gasoline and are run until the gasoline is exhausted. Αt the completion of the experiment the average mileage for each car is computed. Calculations with the data of this one experiment gave a mean of = 18.5 km per litre and a standard deviation of s = 2.2 km per litre for the 25 cars. Long-term experience with cars of the same kind that were used before when no additive was employed indicates that, µ = 18.0 and σ = 2.0 km per litre.

Assuming that the additive had no effect on mileage, answer the following questions:

1.1 Determine the probability accuracy of as an estimate of µ. What is the actual accuracy? Is the sample estimate compatible with what was to be expected by theory?

1.2 Find how large an experiment should have been conducted if one wished to be certain with a probability of 0.95 that the estimate would not be in error by more than ½ km per litre.

1.3 Find a 95 percent confidence interval for µ. Does this interval actually contain µ?

1.4 Dropping the assumption that the additive had nο effect οn either the mean or variance, use Student's t variable to find a 95 percent confidence interval for µ.

Exercise 2:

Answer the following questions:

2.1. Does the accuracy of the sample depend on the size of the population or on the size of the sample?

2.2. If you double the sample size, how much more accurate will your sample become?

2.3. If you decide to accept the null hypothesis, can you be sure that it is really true?

2.4. If you decide to reject the null hypothesis, can you be sure that it is really false?

2.5. Why is it necessary to choose a test statistic, whose distribution is known if the null hypothesis is true?

2.6. How much you have to increase the sample size in order to reduce the confidence interval width by one-half?

Exercise 3:

The Chamber of Commerce and Industry in a recent publication suggests that Small and Medium Enterprises (SME) borrow mainly for working capital and not for investing in equipment. In a survey of 763 SME who recently took bank loans, 229 said that they use the loans for buying new machinery equipment, 435 for working capital, and only 99 for other purposes (letter of credits, etc).

3.1 Set up a 95% confidence interval estimate of the population proportion of SME who borrow for buying new machinery equipment.

3.2 Set up a 95% confidence interval estimate of the population proportion of SME who take loans for working capital.

3.3 Last year in a program financed by the European Union called “Financing SME”, 125 thousand applications for bank loans were approved. Set up a 95% confidence interval estimate of the population number of SME who applied for loans which will be used for working capital.

Exercise 4:

The director of manufacturing at a clothing factory needs to determine whether a new machine is producing a particular type of cloth according to the manufacturer’s specifications, which indicate that the cloth should have a mean breaking strength of at least 70 pounds and a standard deviation of 3.5 pounds. The director is concerned that if the mean breaking strength is actually less than 70 pounds, the company will face too many lawsuits. A sample of 49 pieces of cloth reveals a sample mean of 69.3 pounds.

4.1 State the null and alternative hypotheses.

4.2 At the 0.05 level of significance, is there evidence that the mean breaking strength is less than 70 pounds?

4.3 What will be your answer in 4.2 if you use a 0.01 level of significance?

4.4 What will be your answer in 4.2 if the standard deviation is 4.5 pounds?

(NOTE: Try 4.2, 4.3, 4.4 with sample mean = 69.1 pounds)

Exercise 5:

Feta cheese is placed in packages of nominal weight 500g. The actual weight delivered to a package is normally distributed about the set weight with a standard deviation of 6g.

Calculate the proportion of packages containing

5.1 Less than 485g

5.2 More than 540g

5.3 Between 490g and 520g

5.4 Between what limits of weight lies the middle 95%.