Posted: September 6th, 2015

Quantitative Methods for Business.

Question 1

• You are the manager of a toy company. The company takes its name from your email id (“email id Toy Company” and you insert your own email id into that part of the company name). Your company manufactures a variety of soft toys and you are trying to improve the efficiency of your operations. You wish to adhere to an annual operating budget and to develop operating plans using quantitative methods wherever possible. The overall budget for the year for production of two different soft toys is $500,000. The soft toys are koalas and kangaroos. The budget is intended to pay for labour and materials. Processing requirements for the two toys, on a per unit basis, are shown in the table.

 

Product	Grey cloth	White cloth	Stuffing	Labour
Koala	1.5 metres	0.5 metres	350gms	1.75 hours
Kangaroo	1.0 metres	1.0 metres	400gms	1.50 hours

 

The company has a contractual agreement to produce a minimum of 3,000 toys per year. There must be at least 200 of each toy produced.

 

The production costs for the toys are:

Material	Cost
Labour/hour	$9.50
Grey material/metre	$5.50
White material/metre	$8.50
Stuffing/100gms	$0.35

 

The company can only get a maximum of 5,000 metres of each material.

You wish to minimise the number of labour hours required to meet your contractual obligation.

       Give the full mathematical model for this problem.

1. What are the decision variables for this problem?

 

1. Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximized or minimized? Explain briefly.

 

1. (What is the limited resource in this problem? Using decision variables from part (a), formulate the constraint for this resource.

 

1. What other constraints are relevant to this problem? Using the decision variables from part (a), formulate those constraints.

 

1. Use SOLVER to produce the answer and sensitivity reports required to answer the following questions.

 

1. What is the optimal number of toys?

 

2. The amount of labour hours required to meet the contractual agreement

 

3. How much of the budget has been spent?
4. The manager realises that labour per unit is actually the same for koalas as it is for kangaroos. Does this change the optimum number of toys of each kind to be produced. Explain how you know
5. The company Accountant believes that that amount labour required to make a kangaroo has changed to 1.7 hours. Discuss the impact that this change in the number of hours will have on the minimum value.
6. The Accountant further recommends that the budget be increased to $550,000. What impact if any will this change have on the minimum value?

2.You are asked to analyze the results of a group of QMB students. You decide to summarise their assignment 2 and exam marks and comment on them. The data is found in the file QMB_Results.xlsx

• You decide to describe the shapes (symmetry, modality, outliers) of the distribution for both assignment 2 and Exam marks. To do this you use EXCEL to produce histograms and appropriate summary measures. You then present your findings

(bWhether an observation is an outlier is a matter of judgement. One rule commonly used for identifying outliers is the so-called 1.5 × IQR rule. An observation is suspected to be an outlier if it is more than 1.5 × IQR below the first quartile Q1 or 1.5*IQR above the third quartile Q3.

Apply this rule to both distributions and identify suspected outliers (if any) by using the calculated amounts.

(c)You need to discuss which measures of location and dispersion should be used to describe these distributions? Give (brief) reasons. For each distribution you state which measures should be used giving reasons for your choices.

(d)       You then write down and interpret the values of the summary measures you have chosen

 

Question 3

You then decide to investigate whether there is a linear relationship between the assignment 2 mark received and exam mark using QMB_RESULTS.xlsx. You wish to predict a person’s exam mark based upon their assignment 2 mark.

a.You produce a regression plot that includes the regression equation and the coefficient of determination.

1. You then answer the following questions
2. Which is the independent and which is the dependent variable?
3. What is the value of the slope for this equation? Interpret what this value means.

• What is value of the intercept for this equation? Interpret what the value means

1. Calculate the coefficients of determination for this data set and explain what the value means.
2. Using the equation predict the exam mark when assignment 2 mark is 40%.
3. Are you confident about your prediction? Give reasons for your answer.

• Should any marks be removed from the data sets?

 

Question 4

The following contingency table contains a breakdown of the age and gender of a sample of 100 South Australian general practitioners in 2013-2014.

		Age
		<35	35-44	45-54	>55	Total
Gender	Male	5	15	20	24	64
	Female	5	13	12	6	36
	Total	10	28	32	30	100

 

(a)   Produce a stacked percentage bar chart using the data in this contingency table

(b)  What is the probability that one randomly selected general practitioner is greater than 55 years old?

(cWhat is the probability that one randomly selected general practitioner is both a woman and 35-44 years

What is the probability that one randomly selected general practitioner is a woman or >55 years old?

 

(eWhat is the probability that one randomly selected general practitioner is a woman given she is 45-54 years old?

 

(f) Are the events Male and Age > 55 dependent? Give reasons for your answer.

 

 

 

 

 

 

 

 

 

Question 5

Suppose the average weekly earnings of a production worker in 2014 was $824.20. A researcher randomly selects 54 production workers and obtains a representative earnings statement for one week from each worker.

 

(a)  If the current average weekly earnings of the workers keeps the same as that of 2014, find the probability that the researcher gets a sample average of greater than $850 out of the 54 workers? Assume that the population standard deviation is $33.90. Please include a diagram to illustrate your answer

(b) Assuming the resulting sample average of the 54 randomly selected workers is $832.69, construct the 90% confidence interval of the mean weekly earnings of the production worker today. Assume that the population standard deviation is $33.90. Interpret the meaning of the confidence interval.
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