1

COB291, Management Science, Test 1, Spring 2009 KEY

1. You are working for Caveat Emptor, Inc., a home inspection service that provides prospective home buyers with a thorough assessment of the major systems in a house prior to the execution of the purchase contract. You are asked to forecast the average monthly heating cost with the information given below.

Table 1. Average Monthly Heating Costs and Forecasts

(1) (5 pts) What is the two – period simple moving average forecast (SMA2) for the average heating cost of the home in the month 19? (must show the equation and the process for your answer) SMA(2) for Month 18= F18 = (A16 + A 17) / 2 = (71+206)/2 = 138.5 SMA(2) for Month 19 = F19 = (A17 + F18)/2 = (206 + 138.5) /2= 172.25

Month AHC SMA2

15 138 211.00

16 71 186.50

17 206 104.50

18 138.50

19 172.25

������ � �� � � �⋯…���

=AVERAGE(B7:B8)

(2) What is the three – period weighted moving average forecast (WMA3) for the average heating cost of the home in the month 19 using the weights 0.5, 0.4 and 0.1 with the largest weight for the most recent month and the smallest weight for the remote month?

a. (6 pts.) What is the equation to be used? As in the Table 1, w1=.5, w2=.4, and w3=.1 WMA(3) = F19 = w1*A18 +w2*A17+w3*A16

b. (6 pts.) What is the Excel@ formula to be used for in Cell D9?

=SUMPRODUCT($H$2:$H$4,B6:B8)

c. (3 pts.) What is the answer for it? (must show details by putting numbers in the equation and derive the answer)

2

WMA(3) for Month 18 = F18 = .5*206+.4*71+.1*138=145.2 WMA(3) for Month 19 = F19 = .5*145.2 + .4*206+.1*71=162.1

(3) What is the Exponential smooth forecast for Month 19 using α value of 0.3? a. (5 pts.) What is the equation to be used? (only equation, no need to put any numbers in the

equation)

EXP(t) = Ft = Ft-1+α(At-1-Ft-1), and for Month 19 F19 =F18+ α (A18-F18) b. (5 pts.) What is the Excel@ formula to be used for it in Cell E9?

=E8+$H$7*(B8-E8)

c. (3 pts.) What is the answer for it? (must show details by putting numbers in the equation and

derive the answer) F18 =F17 + α *(A17-F17) = 127.117+.3*(206-127.117)=150.782 F19=F18+ α*(A18-F18) = F18+ α*(F18-F18) = F18 = 150.78 because assume A18=F18 Table 2. MSE and MAD(or MAE) of SMA2

3

(4) As in Table 2 above, how to compute the Mean Squared Error (MSE) for the two – period simple moving average forecast (SMA2) for months 4 to 10?

a. (6 pts.) What is the equation to be used? (only equation, no need to put any numbers in the equation)

Mean Squared Error (MSE): =

��� � 1�� ��� − ����

�

���� ���

�

b. (6 pts.) What is the Excel@ formula to be used for it in Cell C10? =SUMXMY2(B2:B8,C2:C8)/COUNT(B2:B8)

(5) Use the data of the average heating costs and the two – period simple moving average forecasts in Table 2, assume the column F in Table 2 is used to compute the absolute forecasting error from months 9 to 10, answer the following questions:

a. (6 pts) What is the equation to be used to compute Mean Absolute Deviation (MAD or MAE)?

Mean Absolute Error(or Deviation) (MAE or MAD) = (|A1-F1|+|A2-F2| +… …+|An-Fn|) / n

���/��� � 1�� |�� − ���|

�

���

b. (6 pts) What is the MAD for months 9 to 10? (must show details and keep at least three decimal points)

8 AHC SMA2 Error ABSErr

9 42 262.00 -220.00 220

10 90 103.00 -13.00 13

11

12

MAD= 116.5

MAE or MAD = (220+13)/2 = 116.5

-----END of Monthly Heating Cost Forecasting with SMA, WMA and EXP -----------

2. Table 3 below shows the Summary Output for using regression analysis to forecast the average heating costs as the function of the square footage. Answer the following questions based on the information provided in Table 3.

Please note, the Tables 4 to 11 are on a separate sheet of pdf file.

4

Table 3. Simple Linear Regression Analysis for Average Heating Costs

5

(1) (10 pts.) What is the general linear model to be used to model the linear trend for the average heating cost as the function of square footage of the house?

���� � ! � ��"��#� where AHCi is the average heating cost with the square footage of the house i, SFi. β0 is the Y intercept, or the AHC when SFi = 0. β1 is the slope of the regression line drawn with SFi as independent variable and AHCi as dependent variable (Y). It shows the marginal change (increase or decrease) in AHC when the variable SFi changes one unit (increase or decrease).

(2) Answer the following two questions:

a. (2 pts) What is the estimated value b1 for β1?, 0.195

b. (4 pts) How do you interpret the meaning of b1? b1 =0.195 is the slope of the regression line drawn with SFi as independent variable and AHCi as dependent variable (Y). It shows the marginal change (increase or decrease) of 0.195 in AHC when the variable SFi changes one unit (increase or decrease).

(3) You are asked to predict the average heating cost for a house with the size of 2,200 square footage: a. (6 pts.) What is the Excel@ formula of the estimated regression function in Cell D44 of Table 3?

=$B$17+$B$18*B44

b. (3 pts.) What is the predicted average heating cost for a house with the size of 2,200 square footage? (show details)

2200

245.67668

(4) (3pts) You are asked to predict the average heating cost for a house with the size of 500 square footage,

discuss how you would approach the problem and why?

The 500 square footage is outside of the original range of the independent variable for regression model. There is no statistical base to let you do the forecasting unless data collected with square footage of 500 or less included in the data for the regression model.

6

7

(5) Tables 4 to 11show the Summary Output tables for the regression model to predict the average heating cost (Y) as a function of the average outside temperature in winter (X1), the amount of attic insulation in the

house (X2), the age of the furnace in the house (X3), and the size of the house measured in square feet

(X4). If the company wanted to build a regression model using only THREE independent variables out of

the four to predict the average heating cost of these houses, what variables should be used? What is the

criterion used in your choice?

a. (5 pts) What variables should be used and why? Show the detailed reasoning process to your

conclusion in each step of the whole process.

Two major steps with different criteria to be used in each.

1) Use p-value less than the significance level alpha (α) of 5% to screen out any model

(three variable combination) that has the p-value >= 5%. In this case, all of the four

models failed, thus could not be used without future study.

2) Each of the models without their intercepts is also obtained. The p-value tests are

applied again as the results showing in Tables 5, 7, 9 and 11. Now, only all of the p-

values for b1, b2 and b3 in Table 9 are all less than 5%. Thus this is the only valid model

for the case. The three independent variables are square footage, average outside

temp and attic insulation.

b. (5 pts) What is the criterion used in your choice in each step of your reasoning process?

If multiple models are there after the p-value tests, then we would use the highest R2 to determine which one to be used in the final model. If it is asked for the best model without considering the constraint of the THREE independent variables, then the highest adjusted R2 should be used to find the best model with the smallest number of independent variables possible.

------- END of Heating Cost Regression Analysis -------

3. One hundred doctors were asked this hypothetical question: In a routine examination, you find a lump in a

female patient’s breast. In your experience, only 1 out of 100 such lumps turns out to be malignant; but to be

safe, you order a mammogram X ray. If the lump is malignant, there is a 0.80 probability that the mammogram

will identify it as malignant; if the lump is benign, there is a 0.90 probability that the mammogram will identify it

as benign. In this particular case, the mammogram identifies the lump as malignant. In light of these

mammogram results, what is your estimate of the probability that this lump is malignant? You may use the

following table to facilitate your answers. Please keep 3-decimal points of accuracy and show details for each type of probabilities with equations and appropriate notations.

8

Table 12. Probabilities

Please note these notations might be used for the answers: P(M), P(B), P(IdM|M), P(IdB|M), P(IdM|B), P(idB|B), P(IdM & M), P(IdM & B), P(IdB & M), P(IdB & B), P(IdM), P(IdB)

(1) (3 pt) Based on your results above, what is the probability that a randomly selected patient has a benign lump? (must show details with equations and appropriate notations) P(B) = 99%

(2) (5 pts.) Based on your results above, among the patients with malignant lumps, what is the probability that a randomly selected patient is identified as malignant? (must show details with equations and appropriate notations) P(IdM|M) = P(IdM & M) / P(M) = .008/.01 = 80%

(3) (5 pts ) Based on your results above, what is the probability that a randomly selected patient has either a

benign lump or is identified as having a benign lump? (must show details with equations and appropriate notations) P(IdB OR B) = P(IdB) + P(B) – P(IdB & B) = 0.893+0.99-0.891=0.992=99.2%

(4) (5 pts.) Based on your results above, among those patients identified as having malignant lumps, what is

the probability that a randomly selected patient has malignant lump? (must show details with equations and appropriate notations)

=P(M|IdM) = P(IdM & M) / P(IdM) = 0.008/0.107 =0.075=7.5%

(5) (5 pts) For a patient with a lump in her breast, the mammogram identifies the lump as malignant, of the

100 doctors surveyed, 95 gave probability of around 75%. What is the correct probability? You may not

need much computation if you have answered questions above. The claim of those doctors is wrong. They must have confused the P(M | IdM) = .075 and P(IdM | M)=.80. As we see from the two answers to these two questions, they are significantly different.

--------- END of ProbabilityProblem -------

9

4. The Gorman Manufacturing Company must decide whether to make a component part P at its own plant or to purchase the component part P from a supplier. The resulting profit is dependent upon the demand for the product. The following payoff table shows the projected profit (in thousands of dollars).

Table 13. Payoff Table

Based on their experience, the probabilities of low and high demands are 0.10 and 0.50 respectively.

(1) You must show details with appropriate equations and terms and keep 3 decimal points.

a. (6 pts) Use the payoff table and other information available to compute the expected monetary values (EMV) and Max EMV.

b. (8 pts) Construct the decision tree to find the expected monetary values (EMV) and max EMV for

the problem. Each branch and note of the tree should be clearly labeled with its name, payoff, probability, and any results derived from its subsequent notes and branches. You could refer to existing results either given or derived for previous questions without redo them again.

10

c. (4 pts.) What is the maximum or optimal expected profit (EMV) = ___64_______________? (You must show details. You could refer to existing results either given or derived for previous questions without redo them again.)

d. (4 pts.) What is the optimal strategy for the vendor and why? The optimal strategy for the vendor is to Make P with Max EMV of 64. If High Demand, the vendor could get 100, medDemand, 40, but lowDemand -20, with the Max EMV of 64.

(2). What are the Expected Value with Perfect Information (EVwPI) and the Expected Value of Perfect Information (EVPI)?

a. (8 pts) Construct the decision tree to find the Expected Value with Perfect Information (EVwPI) for the problem. Each branch and note of the tree should be clearly labeled with its name, payoff, probability, and any results derived from its subsequent notes and branches. You could refer to existing results either given or derived for previous questions without redo them again.

11

b. (3 pts.) What is the Expected Value of Perfect Information (EVPI) for the problem? (You must show details)

EVPI = EVwPI – EVwoPI = EVwPI – Max EMV = 69 -64 =5

(3) Given the portion of the expanded decision tree with a potential market research for the situation below,

Assume the Company is asked to pay $15 for a reliable market research report. You should use ONLY the information on this expanded decision tree. Assume EVwPI is $110. For your convenience, you may see: 62.06 = -35 * 0.213 + 25 * 0.34 + 85 * 0.6387.

12

a. (10 pts) Compute the Expected Value with Sample Information (EVwSI) and the Expected Value

of Sample Information (EVSI) for the problem. You must show details. EVwSI = 90 + 15 = 105 Max EMV = EVwoSI = 100 EVSI = 5

b. (3 pts) Compute the efficiency of the sample information and discuss your findings. You must

show details. EVwPI = 110 EVwoPI =Max EMV = 100 EVPI = 110 – 100 = 10 E=EVSI / EVPI=5/10=50%

c. (3 pts) What choice would you recommend to the Company for the whole situation (with or without the market research) and why?

E is reasonably large, however, the $15 of fees charged by the Consulting Company is too high relative to the potential benefit of only (EVSI = $5), therefore, should NOT use the search.

(4) With the Company information in Problem 4 reproduced here for convenience, suppose the following data are also available.

Table 13. Payoff Table

Based on their experience, the probabilities of low and high demands are 0.10 and 0.50 respectively.

13

a. (5 pts) Use the Payoff table, the probabilities, etc, to finish a sketch of Market Study portion of the

expanded decision tree on the next page. It should include each branch and note as required.

a. (3 pts) Label appropriately the path of Market Research, High Demand, UnFavorable Report, and to Purchase P. Each branch and note of this portion of the tree should be clearly labeled with its name, payoff, and probability.

14