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15.060 Data, Models, and Decisions Homework 1

Due: MONDAY, September 15, 2014

1. The number of hits per hour at the website http://thenextbigthing.com is a random variable. The probability distribution of the number of hits per hour at this site is shown in Table 2.34.

Hits per Hour Probability xi pi

0 0.03 1 0.09 2 0.24 3 0.27 4 0.25 5 0.09 6 0.03

Table 2.34: Probability distribution of hits per hour at the website http://thenextbigthing.com .

(a) For a given hour, what is the probability that the number of hits will be between 3 and 5 ?

(b) Compute the mean, the variance, and the standard deviation of the number of hits per hour.

2. It is estimated that 70% of all visitors to a given website are college students. The remaining 30% are either college graduates or have not attended college. Suppose the website had twelve visitors in the last two hours.

(a) Let X be the number of visitors (among the twelve) who are college students. What assumptions need to be satised in order for X to obey a binomial distribution?

(b) Modeling X as a binomially distributed random variable, what is n and what is p in this case?

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(c) What is the probability that exactly eight of the twelve visitors are college students?

(d) What is the expected number of visitors (of the twelve) who are college students? What is the standard deviation of the number of visitors who are college students?

3. The website http://MEGA-Retail.com features a huge variety of tness products for upscale consumers. These consumers are segmented into smartphone consumers and desktop consumers depending on whether or not the majority of their purchases are conducted via their smartphones or a less-mobile device such as a desktop or laptop computer. Indeed, 38% of the MEGA-Retail.coms consumer base are smartphone consumers, with the remaining 62% being desktop consumers. The Consumer Fraud Division of the Department of Justice has begun a preliminary investigation into the online practices of MEGA-Retail.com after a newspaper article exposed complaints of padded charges. Their rst task is to choose a random sample of 20 customers for interviews and data collection regarding their transactions with MEGA-Retail.com.

(a) What is the probability that 7 of the 20 randomly chosen consumers will be smartphone consumers?

(b) What is the probability that between 12 and 14 of the 20 randomly chosen consumers will be desktop consumers?

(c) What is the probability that 3 or fewer of the randomly chosen consumers will be desktop consumers?

4. You have made it to the nal round of a game show. The announcer asks you the nal multiple choice question, which has four possible answers: (a), (b), (c), or (d). If you answer the question correctly, you win $1,000,000. After listening to the question, you realize you are not sure of the answer! Rather, you think the answer is either choice (a) with probability 60%, or choice (b) with probability 40%. You know for sure the answer is not choices (c) or (d). You have three options at this point:

Choose not to answer the question. In this case, you walk away with $500,000. Select an answer (either (a) or (b)). If you are correct, you win the $1,000,000. If you are incorrect, you only win $32,000.

Phone a friend. In this option, the announcer allows you to phone your friend and ask her/him for help. After listening to your friends response, you then will need to answer the question. You know that:

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Given that the correct answer is (a), your friend will say a with probability 80%.

Given that the correct answer is (b), your friend will say b with probability 80%.

(a) What is the probability your friend says the answer is a? Hint: It may help to make a probability table.

(b) What is the probability that the true answer is b given that your friend says it is b?

(c) Make a decision tree to nd your best strategy. In words, what should your strategy be? How much money do you expect to win?

5. A medical test for malaria is subject to some error. Given a person who has malaria, the probability that the test will fail to reveal the malaria is 0.06. Given a person who does not have malaria, the test will correctly identify that the person does not have malaria with probability 0.91.

(a) If someone has malaria, what is the probability that the test will identify that person as having malaria?

(b) Suppose three unrelated individuals who are not infected with malaria take the test. What is the probability that at least one of the three individuals will be identied by the test as having malaria?

(c) In a particular area of a particular country, 20% of the population suers from malaria. If a random person from this area is tested and his test results indicate that he has malaria, what is the chance that he actually has malaria?

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Due: MONDAY, September 22, 2014

1. The amount of yogurt sold by Clover each day (for their lovely yogurt parfaits) obeys a continuous uniform distribution between 400 ounces and 600 ounces.

(a) What is the probability that tomorrow Clover will sell between 450 and 500 ounces of yogurt?

(b) What is the probability that tomorrow Clover will sell between 300 and 350 ounces of yogurt?

2. Assume that the net revenue per day for a certain e-store is approximately obeys a Normal distribution with mean = $10,000 and standard deviation = $6,500.

(a) What is the probability that tomorrows net revenue will exceed $9,500?

(b) Suppose that the probability that todays net revenue exceeds the value b is exactly 0:60. What is the value of b?

(c) What proportion of the days does the store have negative net revenue?

3. Your company melts bars of gold to produce liquid gold. The daily production of melted gold approximately obeys a Normal distribution with mean = 50 tons and standard deviation = 10 tons. Compute the quantiles of the daily production of melted gold. More specically:

(a) If the probability that daily production is less than a tons is 0.25, then what is a?

(b) If the probability that daily production is at least b tons is 0.25, then what is b?

(c) If the probability that daily production is at least c tons is 0.50, then what is c?

And here is a slightly more challenging question:

(d) What is the probability that daily production will be either more than 70 tons or less than 30 tons? (In other words, what is the probability that our Normally distributed random variable will be least 2 standard deviations away from its mean?)

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4. JoAnne has gathered data on stock prices and rates of return for companies A and B, and has converted this data into the joint probability distribution of rates of return for the two companies shown in Table 1.

Probability (pi)

Annual Rate of Return of Asset A (%)

(xi)

Annual Rate of Return of Asset B (%)

(yi)

0.04 5.1 6.9 0.09 5.3 6.1 0.16 7.8 8.3 0.19 8.2 9.2 0.17 7.7 7.9 0.15 9.0 7.1 0.10 10.7 9.3 0.10 11.1 11.4

Table 1: Joint probability distribution of the rates of return of companies A and B.

(a) Compute the mean, variance, and the standard deviation of the rate of return of company A.

(b) Compute the mean, variance, and the standard deviation of the rate of return of company B.

(c) Compute the covariance and correlation of rates of the return of companies A and B.

5. The website http://HelpingHomeowners.com oers a variety of information to assist homeowners in repair and maintenance. The site features banner display ads of local service companies and local home product companies, for which HelpingHomewners receives ad revenues based on click-thrus. Let X denote the daily number of clickthrus of display ads of homeowner service companies on the site, and let Y denote the daily number of click-thrus of display ads of home product companies on the site. Suppose that X = 254:1 and X = 88:7, and Y = 177:5 and Y = 85:2. Suppose further that CORR(X; Y ) = 0:63.

(a) Suppose that revenue to HelpingHomewners from click-thrus is $0:04/click-thru for service ads and also $0:04/click-thru for product ads. Compute the mean and

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standard deviation of total daily revenues. Compare the relative risk associated with revenues from service companies, revenues from home product companies, and total revenues. What do you observe?

(b) Suppose that revenue to HelpingHomewners from click-thrus is $0:03/click-thru for service ads and is $0:07/click-thru for product ads. Compute the mean and standard deviation of total daily revenues. Compare the relative risk associated with revenues from service companies, revenues from home product companies, and total revenues. What do you observe?

6. You are tasked with constructing a portfolio out of three assets: A, B and C. The annual returns, risk, and correlations between the returns on these assets are given in Table 2. The requirements of the portfolio are: (i) it should be composed of exactly two assets, (ii) the funds should be invested 50% in each asset, and (iii) the expected annual return of the portfolio should be at least 10%.

Asset Annual Expected

Return(%) Risk (%)

Correlation A B C

A B C

13 7 9

13 12 15

-0.20 -0.15

-0.20

0.30

-0.15 0.30

Table 2: Annual returns, risk, and correlation between the returns of assets A, B, and C.

(a) Find the portfolio that satises all of requirements and has the minimum risk.

(b) [Challenge] Can you nd a portfolio with less risk if you can invest any percentage in each of the two assets (while still satisfying the minimum 10% expected return requirement)?

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12345678910111213141516171819202122232425262728293031323334353637383940414243

Company Number Total Compensation ($thousand) Years in current position Change in stock price from previous year (%) Change in Company's sales from Previous Year (%)

580 2 -15 -67 295 0 -12 49

3,067 10 109 -27 2,971 7 44 26 509 1 28 -7 794 4 10 -7

1,564 8 16 -4 809 4 11 50

1,272 7 -21 -20 594 4 16 -24 230 2 -5 64 936 4 8 -58

1,066 5 28 -73 486 4 13 31

1,295 6 34 66 1,844 7 49 -4 988 5 26 55

1,738 7 46 10 1,732 7 46 -5 2,362 8 63 28 1,565 10 12 -36 2,309 7 48 72 2,913 8 7 5 213 0 -18 -16

3,029 11 102 51 1,567 7 42 -7 2,060 9 55 122 247 0 -14 -41 248 2 -17 -35 856 5 23 19

1,989 8 66 76 2,307 8 67 -48 1,409 5 21 64 1,929 7 46 104 308 0 -11 99

2,243 8 60 -12 1,261 5 10 20 203 0 -17 -18

2,403 11 37 27 2,006 6 40 41 1,753 8 -24 -41 1,283 5 21 87 652 3 1 -34


Due: WEDNESDAY, October 1, 2014

1. The spreadsheet EXEC-PAY.xls contains data on the annual compensation of CEOs of 43 publicly-traded corporations, as well the number of years that the executive has been the CEO of the company, the percentage change in the stock price of the company from the previous year, and the percentage change in the companys sales from the previous year.

.(a) Download the spreadsheet EXEC-PAY.xls Then construct and run a regression model to predict the CEO compensation as a function of the three independent variables indicated in the spreadsheet.

(b) What is the R2 of the regression? In your opinion is this value low or high?

(c) What is the standard error of the regression?

(d) What is the regression equation produced by your linear regression model?

(e) What is the predicted CEO compensation for a CEO who has been with her company for 6 years, and for which the companys stock price has increased by 34% in the last year, while the companys sales have decreased by 3% in the last year?

(f) Do you notice anything unusual about your regression model that might make you less condent about the validity of the model?

2. The Sloan Cafe bakes crumpets that are sold only in the afternoons. Suppose Jill Preisig would like to predict the sales of afternoon crumpets in order to improve eciencies in the baking operation. Jill has collected daily data on sales of afternoon crumpets and has run a regression model with the following independent/explanatory variables:

the number of students who park in the Sloan parking lot on the given day the outside temperature, in degrees Fahrenheit, at noon on the given day the number of croissants sold at breakfast in the morning of the given day the number of crumpets sold the previous afternoon of that day the number of cases due in any Core class on that day

Below is the regression model output from Jills model.

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(a) What is the regression equation of Jills model?

(b) What is the R2 of Jills regression? In your opinion is this value low or high?

(c) According to Jills model, what is the prediction of daily crumpet sales if 40 students parked in the Sloan lot that day, the temperature is 65F, the number of croissants sold at breakfast was 40, the number of crumpets sold yesterday was 10, and there was one Core class case due today?

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15.060 Data, Models, and DecisionsFall 2014


Annual Hops Malt Annual InitialBeer Sales (ounces (pounds Advertising Bitterness Investment

Number ($ thousand) per keg) per keg) ($) Scale ($ million)1 4,800 9.0 8.0 180,000 3 1.42 5,100 9.0 8.0 180,000 3 0.63 5,100 8.0 8.0 180,000 3 0.74 1,900 9.0 7.0 140,000 4 2.45 5,300 8.0 8.0 180,000 3 2.36 2,000 12.0 7.0 140,000 5 1.07 3,400 6.0 7.0 170,000 4 1.58 2,700 8.0 6.0 180,000 4 1.89 2,100 12.0 7.0 140,000 5 0.710 2,600 5.0 6.0 180,000 4 1.311 3,600 10.0 7.0 170,000 5 1.912 2,650 9.0 6.0 180,000 5 0.513 2,200 6.0 7.0 140,000 3 2.114 3,750 12.0 7.0 170,000 6 2.115 2,800 7.0 6.0 180,000 5 2.416 4,600 6.0 8.0 160,000 2 2.017 3,700 8.0 7.0 170,000 4 1.718 4,650 11.0 8.0 160,000 4 0.719 5,900 11.0 8.0 180,000 1 1.620 5,900 12.0 8.0 180,000 2 0.921 4,700 6.0 8.0 160,000 2 1.122 3,800 6.0 7.0 170,000 3 1.123 3,900 12.0 7.0 170,000 6 1.124 4,800 7.0 8.0 160,000 3 2.525 6,050 11.0 8.0 180,000 2 1.726 1,750 13.0 6.0 150,000 7 2.327 5,800 8.0 8.0 180,000 4 0.528 1,600 5.0 6.0 150,000 4 0.829 1,700 10.0 6.0 150,000 6 0.830 1,800 14.0 6.0 150,000 7 0.931 4,200 6.0 7.0 170,000 3 2.432 4,100 6.0 7.0 170,000 4 1.433 6,200 9.0 8.0 180,000 3 2.134 2,400 13.0 7.0 130,000 6 1.435 5,400 17.0 7.0 190,000 9 2.236 2,300 5.0 7.0 130,000 4 1.237 2,500 12.0 7.0 130,000 6 0.938 2,550 12.0 7.0 130,000 6 1.939 6,000 9.0 8.0 170,000 2 1.740 5,500 11.0 7.0 190,000 6 2.241 5,800 8.0 8.0 170,000 4 0.642 6,000 8.0 8.0 170,000 3 2.243 6,100 8.0 8.0 170,000 2 2.144 5,600 6.0 7.0 190,000 4 1.545 5,700 17.0 7.0 190,000 9 1.346 6,250 8.0 8.0 170,000 2 1.247 4,900 5.0 7.0 170,000 3 1.548 5,000 5.0 7.0 170,000 2 1.649 5,100 8.0 7.0 170,000 3 1.450 5,200 8.0 7.0 170,000 1 0.5


Due: WEDNESDAY, October 29, 2014

1. Exercise 7.2 of the DMD textbook. Answer questions (a), (b), and (c).

2. Exercise 7.7 of the DMD textbook. Answer question (a) only, where you should write down (or type) your formulation of the optimization problem. You are not expected to solve this linear optimization problem, only to construct the formulation of the model.

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Due: MONDAY, November 3, 2014

1. Read and analyze the Filatoi Riuniti case at the end of Chapter 7 of the text. Construct a linear optimization model. Write up and hand in your answers to questions (a), (b), (c), (f), (g), (h), and (i), at the end of the case.

2. Exercise 7.7, continuation.

(i). Answer question (b) of Exercise 7.7 by constructing and solving your linear optimization model on your computer. For your added convenience, we have provided a spreadsheet that contains Table 7.38. This spreadsheet contains other data needed in this exercise as well.

(ii). What is the value to Jordan Alloy Corporation of an extra ton per month of copper?

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Metal W X Y Z Availability

(tons/month)

Aluminum 0.30 0.40 0.10 0.15 600

Copper 0.30 0.10 0.25 0.40 400

Magnesium 0.40 0.50 0.65 0.45 800

Contribution to Earnings ($/ton) 35.00 47.00 60.00 140.00

Data for the Dream Team

Average Average AveragePlayer Rebounds Assists Height Points Defensive Ability

Number Per Game Per Game (feet and inches) Per Game (scale from 0 to 10)1 1 7 5' 11" 10 102 2 14 6' 0" 14 93 3 12 6' 4" 19 84 4 4 6' 0" 18 65 5 9 6' 3" 20 86 7 6 6' 5" 21 107 7 8 6' 8" 23 108 4 2 6' 5" 13 59 8 2 6' 10" 17 810 5 5 6' 4" 25 811 10 6 6' 10" 20 912 8 8 6' 9" 30 1013 10 2 7' 3" 24 914 9 5 6' 10" 15 715 6 3 6' 10" 17 616 16 2 6' 9" 3 617 11 1 7' 4" 27 918 12 5 7' 2" 26 1019 11 1 7' 3" 21 920 9 1 7' 0" 14 8


Due: MONDAY, November 24, 2014

1. Exercise 9.3, parts (a) and (b).

2. Solve the optimization problem described in \National Basketball Dream Team" on pages 476-478 of the textbook. Answer questions (a) and (b). You do NOT need to answer question (c). The spreadsheet for this question is provided.

3. Read the case International Industries, Inc. at the end of Chapter 9, and be prepared to discuss the case in class on Monday. You do NOT need to build or solve the model for the case.

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