QUANTITATIVE METHODS FOR BUSINESS

2018-03

Assignment 2: Due Friday 18th January at 11:00 pm

The assignments are to be submitted via the submission link on the Portal. Any submission handed to your teacher in paper form will not be accepted without prior written approval.

Assignments submitted late, without an extension being granted, will attract a penalty of 10% per each calendar day or part thereof beyond the due date and time. Please refer to the Course Information Booklet for the course policy regarding extensions.

Please make sure that the pages of your submission are in the correct order. Please make sure that each page shows your name and the page number.

Your assignment submission must be submitted on-line utilizing Microsoft Word. Poor presentation, including poor layout of formulae and/or explanations of the process, will attract a penalty of 10%. Use of Microsoft Equation Editor 3.0 is encouraged.

For this assignment you must show all your work, including formulae used, explanation of variables, together with a discussion on the process employed. All steps of the process must be shown. Any items omitted will result in loss of marks.

Please remember to complete and attach a cover sheet with a plagiarism declaration. Plagiarism is the use of another person’s ideas (and words) without saying where you got the information. “the taking and using as one’s own the thoughts, writings or inventions of another” The Shorter Oxford English Dictionary on historical principles, Onions CT, third edition. The lack of in text references together with a reference list will, as a minimum, result in zero marks for the section in question. The maximum penalty will be zero marks for the paper. The copying (even partially) of another person’s work will result in both the person who supplied the work to be copied and the person doing the copying receiving zero marks. For further details please see the last page of this assignment.

Please be advised that assignments will be processed through TurnItIn to check for plagiarism. Your submission of the assignment indicates your acceptance to having your paper checked in this way.

Do not rewrite the questions as part of your submission.

Question 1 (Total of 21 marks)

No EXCEL preparation is required for this question

Part A:

Joe Bright, the marketing manager for Mountain Mist soft drink needs to decide how many TV spots and magazine advertisements to run during the next quarter. Each TV spot costs $5,000 and is expected to increase sales by 300,000 cans. Each magazine advertisement costs $2,000 and is expected to increase sales by 500,000 cans. The total soft drink advertising budget for the next quarter is set at $100,000; however Mountain Mist wants to spend no more than $70,000 on TV spots and no more than $50,000 on magazine ads. Mountain Mist earns a profit of 5 cents on each can of soft drink it sells. Joe has decided to use linear programming to find the most profitable mix of advertisements. He asks for your help.

(a) (2 marks) What are the decision variables for this problem?

(b) (4 marks) Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximized or minimized?

(c) (3 marks) What are the constraints relevant to this problem? Using the decision variables from part (a), formulate those constraints.

(d) (2 marks) Give the full mathematical model for this problem.

Part B:

Wen, Yi and Ming are the sole partners and workers in a company that produces fine ceramic figurines. Wen and Yi are each available to work a maximum of 40 hours per week at the company, while Ming is available to work a maximum of 20 hours per week.

The company makes two different types of ceramics: large floor sculptures and small table items. To make an item, Wen assembles the necessary materials and produces the first mould while Yi finalises the process including the curing of each item. Ming is responsible for taking orders and shipping the finished products. The amount of time required for each of these tasks is shown below.

Time required Task Floor sculptures Table figurines

Material and initial cast 6 hours 4 hours

Finalising 8 hours 4 hours

Pack and Ship finished item 3 hours 3 hours

Each floor sculpture built and shipped yields a profit of $310, while each table figurine yields a profit of $210.

Their reports for the Solver solution to the problem are shown below:

Solver Model

Sculptures Figurines Profit

$1,733.33

Total $1,033.33 $700.00

Variables Total Output

Output 3.33 3.336.67

Constraints Total RHS

40.00 40.00

20.00

6 4

8 4

3 3

310 210

Material and initial cast33.33

Finalising40.00

Pack and Ship finished item20.00

Profit / Unit

Microsoft Excel 15.0 Answer Report

Objective Cell (Max)

Cell Name Original Value Final Value

$H$2 Total Profit $1,460.00 $1,733.33

Variable Cells

Cell Name Original Value Final Value Integer

$D$5 Output Sculptures 2.00 3.33 Contin

$E$5 Output Figurines 4.00 3.33 Contin

Constraints

Cell Name

$G$10 Pack and Ship finished item Total

$G$8 Material and initial cast Total

$G$9 Finalising Total

$D$5 Output Sculptures

$E$5 Output Figurines

20.00 $G$10 =$H$10 Binding

33.33 $G$8 =$H$8 Not Binding

40.00 $G$9 =$H$9 Binding

3.33 $D$5 =0 Not Binding

3.33 $E$5 =0 Not Binding Slack

0.00 6.67 0.00

3.33

3.33

Microsoft Excel 15.0 Sensitivity Report

Variable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$D$5 Output Sculptures 3.33 0.00 310.00 110.00 100.00 $E$5 Output Figurines 3.33 0.00 210.00 100.00 55.00

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$G$10 Pack and Ship finished item Total 20.00 36.67 20.00 10.00 5.00 $G$8 Material and initial cast Total 33.33 0.00 40.00 1E+30 6.67 $G$9 Finalising Total 40.00 25.00 40.00 13.33 13.33

(e) (2 marks) Provide the optimal solution to Wen, Yi and Ming.

Which of the EXCEL reports helps you answer this question?

(f) (2 marks) Are there any restrictions on Wen, Yi and Ming producing more ceramics? If so, explain what they are and which report provides that information.

(g) (2 marks) If Wen, Yi and Ming were able to spend more time, let us say 10 hours, in Material and initial cast processes, could they make a larger profit? If so, how much more profit could they make? Which report provides this information?

(h) (2 marks) There is an unusual value ‘1E+30’ in one of the reports. What does this value represent and what does it mean in the context it is shown?

(i) (2 marks) Wen, Yi and Ming enjoy making Figurines more than making Sculptures. If they were to increase the price of their Figurines, what is the maximum amount they could increase their price by and not change the optimal solution?

Please ensure you write properly constructed sentences in response to the above. Single word answers or poorly constructed answers will not receive full marks.

Question 2 (Total of 15

A sample of 500 respondents was selected in a large metropolitan area to determine information concerning consumer behaviour. The following contingency table was obtained:

Gender Male Female Total

Enjoys shopping for clothing

Yes 136 224 360

No 104 36 140

Total 240 260 500

A respondent is chosen at random.

(a) (8 marks) What is the probability that the respondent:

(i) is male;

(ii) does not enjoy shopping for clothing;

(iii) is male and does not enjoy shopping for clothing; (iv) is female or enjoys shopping for clothing.

(b) (3 marks) Assume the respondent chosen enjoys shopping for clothing. What, then, is the probability that the individual is a male?

(c) (4 marks) Are enjoying shopping for clothing and gender of the individual statistically independent? Explain your answer carefully.

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Question 3 (Total of 26

Problems with a telephone line that prevent a customer from receiving or making calls are disconcerting to both the customer and the telephone company. These problems can be of two types: those that are located inside a central office and those located on lines between the central office and the customer’s equipment. The following data represent samples of 20 problems reported to two different offices of a telephone company and the time to clear those problems (in minutes).

Central Office (1) Time to clear problems (minutes)

2.52 0.78 0.93 1.02 1.48 1.60 1.75 2.97 3.93 5.45

0.53 0.80 0.97 1.05 1.48 1.60 2.85 3.10 4.15 6.32

Central Office (2) Time to clear problems (minutes)

0.08 0.52 0.60 0.65 1.10 1.53 1.92 3.30 3.75 2.23

0.10 0.58 0.60 0.72 1.48 1.65 2.10 3.75 4.02 7.55

(a) (8 marks) Based upon groupings of 1 minute intervals use EXCEL to obtain a histogram for each central office location.

(b) (6 marks) Prepare a full Descriptive Statistics profile, including Quartile 1 and Quartile 3, for each central office location. Explain how you prepared your results.

(c) (4 marks) Based on the histograms in (a), briefly describe the shape (symmetry, modality) of the data for each central office location. Do outliers exist in the data sets? Show how you know.

(d) (4 marks) Which measures of location and dispersion should you use to describe the data for each central office location? What are their values? Give a brief explanation of the reason for your choice.

(e) (4 marks) On the basis of your results of (a)-(d), are there any differences between the two central offices? Justify your answer carefully.

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Question 4 (Total of 16

During an orientation event at a large university, incoming students are advised about the cost of textbooks for a typical study period. A sample of 100 students currently enrolled at this university indicates a sample mean cost of $315.40. A population standard deviation is known to be $43.20.

(a) (8 marks) Construct a 95% confidence interval for the true population mean cost of textbooks. Explain what the values you have calculated mean. Provide a diagram (a template is available on the course website) to assist in your explanation.

(b) (3 marks) Does the population cost of textbooks have to be normally distributed here? Explain briefly.

(c) (5 marks) Suppose that the campus newspaper previously claimed that the average cost of textbooks was $300 per study period. If the campus newspaper’s claim were still true, what would be the probability, in a sample of 100 students, of a mean cost of textbooks above $310?

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Question 5 (Total of 22 marks)

The multi-faceted issues of electricity pricing, electricity demand and climate change have been a talking point in Australia for quite some time. As such, you are requested to prepare an analysis of Victorian demand for work days (i.e. Monday to Friday) for Victoria for 2014. The Excel file ‘elecdemand’ contains all of the necessary data for your analysis.

(a) (8 marks) Construct an Ordinary Least Squares Regression (simple Linear Regression) of the data. Ensure you include all necessary calculations and plots in your analysis.

(b) (5 marks) Discuss the appropriateness of the model calculated in (a). Provide full reasoning in support of your conclusion.

(c) (4 marks) Assuming the model is appropriate for use as a linear regression (ignore any conclusion

you have reached in part (b) above), what is the likely demand for electricity in Victoria if the temperature reaches 35o Celsius. Show all calculations, including any appropriate formula(e).

(d) (5 marks) To answer this part of the question you will need to do some research. Please remember the referencing requirements for academic submissions. In the preparation of linear regression models using the appropriate Excel analysis tool, a graph titled ‘Normal Probability Plot’ is provided. What does this plot purport to show and what does it mean in the context of this question?

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

GRAND TOTAL 100 MARKS

2018-03

Assignment 2: Due Friday 18th January at 11:00 pm

The assignments are to be submitted via the submission link on the Portal. Any submission handed to your teacher in paper form will not be accepted without prior written approval.

Assignments submitted late, without an extension being granted, will attract a penalty of 10% per each calendar day or part thereof beyond the due date and time. Please refer to the Course Information Booklet for the course policy regarding extensions.

Please make sure that the pages of your submission are in the correct order. Please make sure that each page shows your name and the page number.

Your assignment submission must be submitted on-line utilizing Microsoft Word. Poor presentation, including poor layout of formulae and/or explanations of the process, will attract a penalty of 10%. Use of Microsoft Equation Editor 3.0 is encouraged.

For this assignment you must show all your work, including formulae used, explanation of variables, together with a discussion on the process employed. All steps of the process must be shown. Any items omitted will result in loss of marks.

Please remember to complete and attach a cover sheet with a plagiarism declaration. Plagiarism is the use of another person’s ideas (and words) without saying where you got the information. “the taking and using as one’s own the thoughts, writings or inventions of another” The Shorter Oxford English Dictionary on historical principles, Onions CT, third edition. The lack of in text references together with a reference list will, as a minimum, result in zero marks for the section in question. The maximum penalty will be zero marks for the paper. The copying (even partially) of another person’s work will result in both the person who supplied the work to be copied and the person doing the copying receiving zero marks. For further details please see the last page of this assignment.

Please be advised that assignments will be processed through TurnItIn to check for plagiarism. Your submission of the assignment indicates your acceptance to having your paper checked in this way.

Do not rewrite the questions as part of your submission.

Question 1 (Total of 21 marks)

No EXCEL preparation is required for this question

Part A:

Joe Bright, the marketing manager for Mountain Mist soft drink needs to decide how many TV spots and magazine advertisements to run during the next quarter. Each TV spot costs $5,000 and is expected to increase sales by 300,000 cans. Each magazine advertisement costs $2,000 and is expected to increase sales by 500,000 cans. The total soft drink advertising budget for the next quarter is set at $100,000; however Mountain Mist wants to spend no more than $70,000 on TV spots and no more than $50,000 on magazine ads. Mountain Mist earns a profit of 5 cents on each can of soft drink it sells. Joe has decided to use linear programming to find the most profitable mix of advertisements. He asks for your help.

(a) (2 marks) What are the decision variables for this problem?

(b) (4 marks) Using decision variables identified in part (a), formulate the objective function for this problem. Is the quantity of interest to be maximized or minimized?

(c) (3 marks) What are the constraints relevant to this problem? Using the decision variables from part (a), formulate those constraints.

(d) (2 marks) Give the full mathematical model for this problem.

Part B:

Wen, Yi and Ming are the sole partners and workers in a company that produces fine ceramic figurines. Wen and Yi are each available to work a maximum of 40 hours per week at the company, while Ming is available to work a maximum of 20 hours per week.

The company makes two different types of ceramics: large floor sculptures and small table items. To make an item, Wen assembles the necessary materials and produces the first mould while Yi finalises the process including the curing of each item. Ming is responsible for taking orders and shipping the finished products. The amount of time required for each of these tasks is shown below.

Time required Task Floor sculptures Table figurines

Material and initial cast 6 hours 4 hours

Finalising 8 hours 4 hours

Pack and Ship finished item 3 hours 3 hours

Each floor sculpture built and shipped yields a profit of $310, while each table figurine yields a profit of $210.

Their reports for the Solver solution to the problem are shown below:

Solver Model

Sculptures Figurines Profit

$1,733.33

Total $1,033.33 $700.00

Variables Total Output

Output 3.33 3.336.67

Constraints Total RHS

40.00 40.00

20.00

6 4

8 4

3 3

310 210

Material and initial cast33.33

Finalising40.00

Pack and Ship finished item20.00

Profit / Unit

Microsoft Excel 15.0 Answer Report

Objective Cell (Max)

Cell Name Original Value Final Value

$H$2 Total Profit $1,460.00 $1,733.33

Variable Cells

Cell Name Original Value Final Value Integer

$D$5 Output Sculptures 2.00 3.33 Contin

$E$5 Output Figurines 4.00 3.33 Contin

Constraints

Cell Name

$G$10 Pack and Ship finished item Total

$G$8 Material and initial cast Total

$G$9 Finalising Total

$D$5 Output Sculptures

$E$5 Output Figurines

20.00 $G$10 =$H$10 Binding

33.33 $G$8 =$H$8 Not Binding

40.00 $G$9 =$H$9 Binding

3.33 $D$5 =0 Not Binding

3.33 $E$5 =0 Not Binding Slack

0.00 6.67 0.00

3.33

3.33

Microsoft Excel 15.0 Sensitivity Report

Variable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$D$5 Output Sculptures 3.33 0.00 310.00 110.00 100.00 $E$5 Output Figurines 3.33 0.00 210.00 100.00 55.00

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$G$10 Pack and Ship finished item Total 20.00 36.67 20.00 10.00 5.00 $G$8 Material and initial cast Total 33.33 0.00 40.00 1E+30 6.67 $G$9 Finalising Total 40.00 25.00 40.00 13.33 13.33

(e) (2 marks) Provide the optimal solution to Wen, Yi and Ming.

Which of the EXCEL reports helps you answer this question?

(f) (2 marks) Are there any restrictions on Wen, Yi and Ming producing more ceramics? If so, explain what they are and which report provides that information.

(g) (2 marks) If Wen, Yi and Ming were able to spend more time, let us say 10 hours, in Material and initial cast processes, could they make a larger profit? If so, how much more profit could they make? Which report provides this information?

(h) (2 marks) There is an unusual value ‘1E+30’ in one of the reports. What does this value represent and what does it mean in the context it is shown?

(i) (2 marks) Wen, Yi and Ming enjoy making Figurines more than making Sculptures. If they were to increase the price of their Figurines, what is the maximum amount they could increase their price by and not change the optimal solution?

Please ensure you write properly constructed sentences in response to the above. Single word answers or poorly constructed answers will not receive full marks.

Question 2 (Total of 15

A sample of 500 respondents was selected in a large metropolitan area to determine information concerning consumer behaviour. The following contingency table was obtained:

Gender Male Female Total

Enjoys shopping for clothing

Yes 136 224 360

No 104 36 140

Total 240 260 500

A respondent is chosen at random.

(a) (8 marks) What is the probability that the respondent:

(i) is male;

(ii) does not enjoy shopping for clothing;

(iii) is male and does not enjoy shopping for clothing; (iv) is female or enjoys shopping for clothing.

(b) (3 marks) Assume the respondent chosen enjoys shopping for clothing. What, then, is the probability that the individual is a male?

(c) (4 marks) Are enjoying shopping for clothing and gender of the individual statistically independent? Explain your answer carefully.

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Question 3 (Total of 26

Problems with a telephone line that prevent a customer from receiving or making calls are disconcerting to both the customer and the telephone company. These problems can be of two types: those that are located inside a central office and those located on lines between the central office and the customer’s equipment. The following data represent samples of 20 problems reported to two different offices of a telephone company and the time to clear those problems (in minutes).

Central Office (1) Time to clear problems (minutes)

2.52 0.78 0.93 1.02 1.48 1.60 1.75 2.97 3.93 5.45

0.53 0.80 0.97 1.05 1.48 1.60 2.85 3.10 4.15 6.32

Central Office (2) Time to clear problems (minutes)

0.08 0.52 0.60 0.65 1.10 1.53 1.92 3.30 3.75 2.23

0.10 0.58 0.60 0.72 1.48 1.65 2.10 3.75 4.02 7.55

(a) (8 marks) Based upon groupings of 1 minute intervals use EXCEL to obtain a histogram for each central office location.

(b) (6 marks) Prepare a full Descriptive Statistics profile, including Quartile 1 and Quartile 3, for each central office location. Explain how you prepared your results.

(c) (4 marks) Based on the histograms in (a), briefly describe the shape (symmetry, modality) of the data for each central office location. Do outliers exist in the data sets? Show how you know.

(d) (4 marks) Which measures of location and dispersion should you use to describe the data for each central office location? What are their values? Give a brief explanation of the reason for your choice.

(e) (4 marks) On the basis of your results of (a)-(d), are there any differences between the two central offices? Justify your answer carefully.

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Question 4 (Total of 16

During an orientation event at a large university, incoming students are advised about the cost of textbooks for a typical study period. A sample of 100 students currently enrolled at this university indicates a sample mean cost of $315.40. A population standard deviation is known to be $43.20.

(a) (8 marks) Construct a 95% confidence interval for the true population mean cost of textbooks. Explain what the values you have calculated mean. Provide a diagram (a template is available on the course website) to assist in your explanation.

(b) (3 marks) Does the population cost of textbooks have to be normally distributed here? Explain briefly.

(c) (5 marks) Suppose that the campus newspaper previously claimed that the average cost of textbooks was $300 per study period. If the campus newspaper’s claim were still true, what would be the probability, in a sample of 100 students, of a mean cost of textbooks above $310?

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

Question 5 (Total of 22 marks)

The multi-faceted issues of electricity pricing, electricity demand and climate change have been a talking point in Australia for quite some time. As such, you are requested to prepare an analysis of Victorian demand for work days (i.e. Monday to Friday) for Victoria for 2014. The Excel file ‘elecdemand’ contains all of the necessary data for your analysis.

(a) (8 marks) Construct an Ordinary Least Squares Regression (simple Linear Regression) of the data. Ensure you include all necessary calculations and plots in your analysis.

(b) (5 marks) Discuss the appropriateness of the model calculated in (a). Provide full reasoning in support of your conclusion.

(c) (4 marks) Assuming the model is appropriate for use as a linear regression (ignore any conclusion

you have reached in part (b) above), what is the likely demand for electricity in Victoria if the temperature reaches 35o Celsius. Show all calculations, including any appropriate formula(e).

(d) (5 marks) To answer this part of the question you will need to do some research. Please remember the referencing requirements for academic submissions. In the preparation of linear regression models using the appropriate Excel analysis tool, a graph titled ‘Normal Probability Plot’ is provided. What does this plot purport to show and what does it mean in the context of this question?

Note: In order to achieve full marks for this question it is essential that you fully explain what you are doing, why you are doing it and the steps involved in providing a final solution. Ensure your answer is not just a set of calculations as 25% of the marks for this question are set aside for your explanation.

GRAND TOTAL 100 MARKS

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