The employee manager of a large company would like to

Question

Question1: Please select the most appropriate answer to each of the questions below.

1. The null hypothesis usually represents:

i. the theory the researcher would like to prove.

ii. the preconceived ideas of the researcher

iii.the perceptions of the sample population

iv.the status quo

v. none of the above

The employee manager of a large company would like to estimate the proportion of full-time employees who prefer adopting plan A of two available health care plans in the forthcoming annual enrollment period.

2. The employee manager believes that if the proportion of employees preferring plan A is less than 2%, then that plan should be dropped. A 95% confidence interval for the proportion of employees preferring plan A yields (2.05%; 4.12%). Based on this interval, what is the manager's decision regarding whether or not to drop plan A?

i.   Drop the plan

ii.  Do not drop the plan.

iii. Inconclusive. Need more information to make a decision

3. The manager is not satisfied with the accuracy of the confidence interval (2.05%; 4.12%) obtained in question 3. What are the options to increase the interval's accuracy (at the same confidence level)?

i.   To increase the sample size

ii.  To reduce the sample size

iii. To use a smaller multiple

iv. To use a larger multiple

v.  Bothi. and iii.

vi. Both i. and iv.

Question 2:

A new provider of ISDN internet access wants to decide whether to commence operations in PG County. It decides to do so only if the proportion of households in PG County that currently do not have high-speed internet access is significantly larger than the nation-wide average. The nation-wide average (for households that do not have high-speed internet access) is known to be 40%.

A survey of 200 randomly selected households in PG County revealed that 96 of them had no high-speed internet access.

1. State the research hypotheses that must be tested by the internet provider in order to decide whether to commence operations in PG County.

i. H₁: The proportion of households in PG County without high-speed internet is less than 48%

ii. H₁: The proportion of households in PG County without high-speed internet is larger than 48%

iii. H₁: The proportion of households in PG County without high-speed internet is different from 40%

iv. H₁: The proportion of households in PG County without high-speed internet is less than 40%

v. H₁: The proportion of households in PG County without high-speed internet is larger than 40%

2. ASSUME that the p-value is 0.0394. At the 5% level of significance, the internet provider's decision (based on the information only from this test) regarding whether or not to commence operations in PG County is

i. Reject the Null Hypothesis: Commence operations in PG County

ii. Do not reject the Null Hypothesis: Do not commence operations in PG County

iii. Inconclusive. Need more information to make decision.

iv. None of the above

3. Insight: Using a 1% level of significance instead of a 5% level of significance would decrease the chances of committing a Type ______________________________________.

i. Type I error, but increase chances of Type II error

ii. Type II error, but increase chances of Type I error

iii. Type III error, but increase chances of Type II error

iv. Type II error, but increase chances of Type III error

4. In the context of this problem, committing a Type I error means to

i. Conclude that the proportion of households in PG County without high-speed internet is larger than 40%, when in reality it is the same as the nation-wide average!

ii. Conclude that the proportion of households in PG County without high-speed internet is not larger than 40%, when in reality it is larger than the nation-wide average!

iii. Conclude that the proportion of households in PG County without high-speed internet is larger than 40%, when in reality it is larger than the nation-wide average!

iv. Conclude that the proportion of households in PG County without high-speed internet is not larger than 40%, when in reality it is the same as the nation-wide average!

 

Question 3- 1.5 points

Ellen Sarne has collected the following data (see table to the right) on the amount of time (in minutes) taken by a tax preparation service to complete client interviews:

Client Number	Time to complete Interview (Minutes)
1	8
2	12
3	26
4	10
5	23
6	21
7	16
8	22
9	18
10	17
11	36
12	9

Using StatTools, she calculates the following information:

	Time to Complete Interview (minutes)
One Variable Summary	interview times
Mean	18.167
Std. Dev.	8.111
Median	17.000
Minimum	8.000
Maximum	36.000
Count	12

You may assume that this data are a representative sample of all client interview times. Furthermore, please assume that the underlying distribution of interview times is approximately normal.

She continues her analysis and constructs a 99% confidence interval, again using StatTools. The resulting output is:

	Time to Complete Interview (minutes)
Conf. Intervals (One-Sample)	interview times
Sample Size	12
Sample Mean	18.167
Sample Std Dev	8.111
Confidence Level (Mean)	99.0%
Degrees of Freedom	11
Lower Limit	10.895
Upper Limit	25.439

a) Based on her confidence interval she concludes:

a. With 99% confidence the sample mean is 18.167

b. With 99% confidence the interview times are between 10.895 and 25.439

c. 99% of the time a random interview time (a random sample of an interview time) will be somewhere between 10.895 and 25.439

d. Both b and c

e. None of the above

b) Ellen Sarne is unhappy with the width of the interval.  Approximately how large a sample size is required to obtain a 99% confidence interval whose accuracy is +/- 4.0 minutes?

Question 4:

A financial analyst has assembled quarterly sales data for The Gap from the first quarter of 1985 through the fourth quarter of 1997 (i.e., for 52 quarters). 

The variable GapSales measures quarterly sales in thousands of dollars. 

The analyst creates a regression model using Log(GapSales) as the response variable and the following predictor variables:

t = serial number of observation (from 1 for Q1-1985, to 52 for Q4-1997)

D1 = 1 if observation is for a first quarter; 0 otherwise

D2 = 1 if observation is for a second quarter; 0 otherwise

D3 = 1 if observation is for a third quarter; 0 otherwise

	Multiple	R-Square	Adjusted	StErr of
Summary	R		R-Square	Estimate
	0.9935	0.9871	0.9860	0.088127721
	Degrees of	Sum of	Mean of	F-Ratio	p-Value
ANOVA Table	Freedom	Squares	Squares
Explained	4	28.00782572	7.001956429	901.5594	< 0.0001
Unexplained	47	0.365025273	0.007766495
	Coefficient	Standard	t-Value	p-Value	Confidence Interval 95%
Regression Table		Error			Lower	Upper
Constant	12.20138559	0.033468905	364.5589	< 0.0001	12.13405484	12.26871634
T	0.046840608	0.000816558	57.3635	< 0.0001	0.045197906	0.04848331
D1	-0.39966044	0.034653229	-11.5331	< 0.0001	-0.46937375	-0.32994714
D2	-0.39545816	0.034605093	-11.4277	< 0.0001	-0.46507463	-0.32584169
D3	-0.17682630	0.034576179	-5.1141	< 0.0001	-0.24638460	-0.10726800

(a) Provide an economic interpretation for the coefficient of D1.

(b) What is the forecast for the second quarter of 1998?  (Specify the units carefully.)

Question 5:

A vendor of a software product claims that their new program/app called Algebra without Tears greatly enhances the understanding of key mathematical concepts among school students. They approached the Anne Arundel County School District with the suggestion that the program be adopted in county schools and integrated into the curriculum. While the program-which runs on PC/Mac/iPad-has a very nice user interface, it is expensive. There is a $100,000 purchase fee, and an annual maintenance license cost of $12,000. The county wants to know what the effect of adopting this program will be on student outcomes. They are aware that some students are already using the program (on personal licenses) and collect data from three schools. The data is described in the following table:

Variable	Description
Score (Dependent Variable)	Aggregate score on math exams over the last year (out of 400)
Income	Family income in dollars
Use	Whether software has been used for at least 12 months (1 if yes/ 0 if no)
School	School attended (identified as A, B, or C)
Age	Student age (in years)

The categorical variable School, is recoded using two dummy variables (School_A = 1 if School = A, and 0 otherwise; School_B = 1 if School = B, and 0 otherwise). The regression output is given below.

		Multiple	R-Square	Adjusted		StErr of
Summary		R		R-Square		Estimate
		0.9887	0.9775	0.9763		11.72487235
		Degrees of	Sum of	Mean of		F-Ratio		p-Value
ANOVA Table		Freedom	Squares	Squares
Explained		5	561525.0126	112305.0025		816.9263		< 0.0001
Unexplained		94	12922.42736	137.4726315
	Coefficient		Standard	t-Value		p-Value		Confidence Interval 95%
Regression Table			Error					Lower		Upper
Constant	16.17241895		9.489894931	1.7042	0.0917		-2.669989896		35.0148278
Income	0.002478046		3.96295E-05	62.5304	< 0.0001		0.002399361		0.002556731
Use	-1.267374235		2.370366484	-0.5347	0.5941		-5.973792455		3.439043985
Age	8.061659242		0.840529787	9.5912	< 0.0001		6.498765637		9.664830485
School_A	-9.054034011		3.320036986	-2.7271	0.0076		-15.64604527		-2.462022748
School_B	-0.092901467		3.706700576	-0.0251	0.9801		-7.452642352		7.266839418

(a) The p-value for the School_A indicates(please circle the most appropriate answer)

• That School A provides good math education
• That School A is a useful predictor
• That School A is statistically different from School B
• That School A is not useful explaining Score, given the other variables in the model

(b) Provide an economical interpretation of the coefficient of Income

(c) Should the school district adopt Algebra without Tearsand integrate it into the curriculum? Explain.

The analyst hypothesizes that the influence of income on test scores declines as income increase. In other words, the magnitude of change in test scores for a $1000 increase in family income is smaller at higher incomes. How would you go about determining if this is true? [Describe the approach you would use (not amenable to a numerical answer).]

Question6:

The Prudential Bank offers a special low interest loan program for small business owners.  The rate is fixed for all loans made under this program, but the bank has the discretion whether or not to approve a particular loan.

Ann, a loan officer at the Bank, is reviewing an application under this program from Bob, a small business owner.  If Bob repays the loan as per the terms of the loan, the Bank will earn a profit of $15,000.  (All earning are reported as present values.)  However, if Bob defaults on the loan, the Bank will lose $10,000.  Based on quick examination of Bob's application, Ann assesses Bob's likelihood of default at 14.5%.

Ann can either issue a quick decision to approve or deny the loan, or send Bob's application to Credit Services Inc., a firm that conducts in-depth credit analysis and issues a rating of "A", "B" or "C" for the prospective borrower.  Each application sent to Credit Services costs the Bank $2,000.  Historical default rates for Credit Services' three ratings categories are shown in the table on the right:

Rating	Default Probability
A	1%
B	10%
C	90%

Ann believes that if she sends Bob's case to Credit Services, there is a 40% chance of Bob getting an A rating, 50% for a B rating, and a 10% chance for a C rating.

a) Ann wants to utilize a decision tree to specify her optimal course of action. The first node in the tree is a chance/decision (circle one) node representing

b) The following figures show a sensitivity analysis on run on the "Cost of Credit Services". Approximately how much are you willing to pay for the Credit Services? (you may assume Ann's decision tree is correct)

a) The optimal strategy for Ann is

i. Not use the Credit Services and not award the loan

ii. Not use the Credit Services and award the loan

iii. To use the Credit Services and only award the loan if the rating comes back A

iv. To use the Credit Services and only award the loan if the rating comes back A or B

v. We need more information (i.e. the full tree) to answer this question.

Question 7:

Big Cheese Co. makes fine artisan cheese in two locations: Vermont and Oregon.  The firm sells its cheese in New York, Washington DC, Los Angeles and San Francisco.  Estimated monthly demand in each city for the next 3 months is shown below:

	Month 1	Month 2	Month 3
NY	500	450	400
DC	200	200	150
LA	400	350	450
SF	500	600	550

Raw material for each unit of cheese costs $150 in Vermont and $200 in Oregon.  Each unit of cheese requires 4 units of labor in Vermont and 3 units of labor in Oregon.  Each unit of labor used costs $50 in Vermont and $60 in Oregon.  Each month, up to 3,200 units of labor are available in Vermont and up to 4,500 units of labor are available in Oregon.

Per unit shipping costs are shown below:

		To
		NY	DC	LA	SF
From	Vermont	$10	$14	$20	$21
	Oregon	$18	$20	$15	$10

Mr. Mozze A. Rella has specified the following optimization formulation to minimize his costs(both production and shipping costs).

The decision variables

OP_month - units of cheese produced in Oregon in a particular month

VP_month - units of cheese produced in Vermont in a particular month

O_j-month - units of cheese shipped from Oregon to j in a particular month

V_j-month - units of cheese shipped from Vermont to j in a particular month

Example: OP₁ would be units of cheese produced in Oregon in month 1 and O_NY-1 would be units of cheese shipped from Oregon to New York in month 1.

The objective

The objective is to minimize the overall costs of production and shipping and can be written as:

Minimize production costs + shipping costs

where

Production costs = ($150 + 4 *$50) (VP₁+VP₂+VP₃) +($200 + 3 *$60 )(OP₁+OP₂+OP₃)

Shipping costs = $10(V_NY-1+ V_NY-2+ V_NY-3)+ $14(V_DC-1+ V_DC-2+ V_DC-3) +$20(V_LA-1+ V_LA-2+ V_LA-3)+ $21(V_SF-1+ V_SF-2+ V_SF-3) + $18(O_NY-1+ O_NY-2+ O_NY-3)+ $20 (O_DC-1+ O_DC-2+ O_DC-3) +$15(O_LA-1+ O_LA-2+ O_LA-3)+ $10(O_SF-1+ O_SF-2+ O_SF-3)

The constraints

There are three key sets of constraints.First, we cannot produce more than we have the capacity to produce.Second, we need to satisfy demand.Third, we cannot ship cheese unless we have producedit.

Capacity:

For each month i, we have the following labor constraints:

4VP_i≤ 3,200

3OP_i≤ 4,500

Demand:

For each marketj in month i, we need to meet demand:

(V_j-i +O_j-i)≥demand in location j in month i

As an example, the demand constraint for NY in month 1 would be:

V_NY-1+O_NY-1≥500

Balance equations:

We can only ship what we have produced, therefore for each month iwe have:

V_NY-i + V_DC-i +V_SF-i +V_LA-i≤VP_i

O_NY-i + O_DC-i +O_SF-i +O_LA-i≤OP_i

Nonnegativity:

All decision variables are non-negative.

Please answer the questions below,all parts a) - c) are independent from each other.

a) Mr. Rella has just realized that he forgot to model his inventory options. In particular,any quantity of cheese that is not shipped to a market is stored at the production site. Storage cost is $10 (per unit per month) in Vermont and $15 in Oregon.  At the beginning of Month 1, there are 300 units in storage at Vermont and 200 units at Oregon. 

Use the following approximation for the average inventory in each month i:

Average inventory in month i = (starting inventory in month i+ending inventory month i)/2

In the space below, specify a formula that calculates the storage costs in month 1 in Vermont. Make sure the formula is fully specified, that is, all quantities in the formula are decision variables, input parameters or intermediate calculations that you also specify.

It is now the start of month 2, and there are 75 units in storage in Vermont. Write a formula that calculates the end of month 2 inventory in Vermont. Make sure the formula is fully specified, that is, all quantities in the formula are decision variables, input parameters or intermediate calculations that you also specify.

a) It is Month 2, and the cows in Vermont are suffering from BSE (a cattle disease), and therefore the supply chain is disrupted. Mr. Rella is confident that he can contract alternative farms to provide the raw materials for Month 3, but for Month 2,only5,000 gallons of raw milk will be available in the Vermont facility. Each unit of cheese requires 6 gallons of milk.

How would you add this restriction to the model?

What can you say about how it would affect the objective value?

 

b) After formulating his problem, Mr. Rella ran the following sensitivity analysis using Solver Table. He was interested in analyzing the availability and costs of labor at his two locations and how they are affecting his cost (the objective). The following exhibits summarize his analysis.

Cost of labor - Sensitivity Analysis

Availability of labor - Sensitivity Analysis

Based on the information above please select the most appropriate answer to each of the questions.

If given the choice between expanding the available labor in Oregon or in Vermont, Mr. Rella should:

i. Expand the labor availability in Oregon

ii. Expand the labor availability in Vermont

iii. There is not enough information to answer the question appropriately

a) Mr. Rellais concerned about the robustness of his supply chain, in particular he wonders how much it would cost him if he required no more than 75% of the supply for any single market to come from a single source (Vermont or Oregon).

Write down a linear constraint that ensures that no more than 75% of the supply for New York in Month 3 comes from Vermont. Make sure the constraint is fully specified, that is, all quantities in the constraints are decision variables, input parameters or intermediate calculations that you also specify.

Question 8:

Egress, Inc. is a small company that designs, produces and sells Ski jackets and other coats. The creative design team has labored for weeks over its new design for the coming winter season. It is now time to decide how many Ski jackets to produce in the production run. Because of the lead times involved, no other production runs will be possible during the season. Predicting sales of Ski jackets months in advance of the selling season can be quite tricky. Egress has been in operation for only 3 years, and its Ski jacket designs were quite successful in two of those years. Based on realized sales from the last 3 years, current economic conditions and professional judgment, twelve Egress employees have independently estimated demand for their new design for the upcoming season.

To assist in the decision on the number of units for the production run, management has gathered the data below. Note that S is the price Egress charges retailers. Any Ski jackets that do not sell during the season can be sold by Egress to discounters for V per jacket. The fixed cost of plant and equipment is F. This cost is incurred regardless of the size of the production run. The variable cost per jacket is C.

Variable production cost per unit (C):	$80
Selling price per unit (S):	$100
Salvage value per unit (V):	$30
Fixed production cost (F):	$100,000

You have been asked to assist the Egress management in deciding on the best quantity for the production run.They are interested in maximizing their mean profit, but at the same time they do not want to be exposed to large downside risk because they do not have large cash reservesas a young company.

You realize that simulation can be quite helpful. In particular, you realize the demand (D) is uncertain and your goal is to pick the best quantity (Q) for the production run. You use the normal distribution to estimate the demand based on the prediction of the employees (and for the purpose of the final you may assume that this assumption is valid).

a) You model the profit scenario when 9,000 Ski jackets are produced. The resulting histogram shows the profit distribution.

a) Please explain the shape of the histogram. In particular: why is there a spike? And what does the $80,000 maximum correspond to?

b) Realizing that Q is in fact a decision variable, you use the RiskSimTable function in the @Risk software to test multiple values of Q, rangingfrom 7,000 to 14,000 in increments of 1,000. A resulting summary trend graph is shown below (Sim#1 corresponds to Q=7,000, Sim#2 corresponds to Q=8,000, etc.).

Please answer the following questions based on the trend graph:

What is the value of Q that maximizes the expected profit?

Does the Q you stated above, equal the mean demand or not? Explain.

What is your recommended quantity (Q) for the production? Explain.