Decision Analysis Problems

2.24 The Boston South Fifth Street Softball League consists of three teams: Mama's Boys, team 1; the Killers, team 2; and the Machos, team 3. Each team plays the other teams just once during the season. The win-loss record for the past 5 years is as follows: Round 1, 2, 3 Mama's Boys X 3 4 The Killers (2) 2 X 1 The Machos (3) 1 4 X Each row represents the number of wins over the past 5 years. Mama's Boys beat the Killers 3 times, beat the Machos 4 times, and so on. (a) What is the probability that the Killers will win every game next year? (b) What is the probability that the Machos will win at least one game next year? (c) What is the probability that Mama's Boys will win exactly one game next year? (d) What is the probability that the Killers will win fewer than two games next year?

(a) 0.08 (b) 0.84 (c) 0.44 (d) 0.92 MB = Mama's Boys, K = the Killers, and M = the Machos <ART FILE="02_24eq01.eps" W="89.571pt" H="19.937pt" XS="100%" YS="100%"/> <ART FILE="02_24eq02.eps" W="92.071pt" H="19.762pt" XS="100%" YS="100%"/> <ART FILE="02_24eq03.eps" W="89.6pt" H="19.312pt" XS="100%" YS="100%"/> <ART FILE="02_24eq06.eps" W="86.28pt" H="20.162pt" XS="100%" YS="100%"/> a. The probability that K will win every game is P = P(K over MB) and P(K over M) = (0.4)(0.2 ) = 0.08 b. The probability that M will win at least one game is P(M over K) + P(M over MB) - P(M over K) ´ P(M over MB) = (0.8) + (0.2) - (0.8)(0.2) = 1 - 0.16 = 0.84 c. The probability is 1. [P(MB over K) and P(M over MB)], or 2. [P(MB over M) and P(K over MB)] P(1) = (0.6)(0.2) = 0.12 P(2) = (0.8)(0.4) = 0.32 Probability = P(1) + P(2) = 0.12 + 0.32 = 0.44 d. Probability = 1 - winning every game = 1 - answer to part (a) = 1 - 0.08 = 0.92

Two states of nature exist for a particular situation: a good economy and a poor economy. An economic study may be performed to obtain more information about which of these will actually occur in the coming year. The study may forecast either a good economy or a poor economy. Currently there is a 60% chance that the economy will be good and a 40% chance that it will be poor. In the past, whenever the economy was good, the economic study predicted it would be good 80% of the time. (The other 20% of the time the prediction was wrong.) In the past, whenever the economy was poor, the economic study predicted it would be poor 90% of the time. (The other 10% of the time the prediction was wrong.) (a) Use Bayes' Theorem and find the following: P good economy prediction of good economy P poor economy prediction of good economy P good economy prediction of poor economy P poor economy prediction of poor economy (b) Suppose the initial (prior) probability of a good economy is 70% (instead of 60%) and the initial probability of a poor economy is 30% (instead of 40%). Find the posterior probabilities in part (a) based on these new values.

(a) 0.923, 0.077, 0.25, 0.75 (b) 0.949, 0.051, 0.341, 0.659

3.50 In the past few years, the traffic problems in Lynn McKell's hometown have gotten worse. Now, Broad Street is congested about half the time. The normal travel time to work for Lynn is only 15 minutes when Broad Street is used and there is no congestion. With congestion, however, it takes Lynn 40 minutes to get to work. If Lynn decides to take the expressway, it will take 30 minutes, regardless of the traffic conditions. Lynn's utility for travel time is U(15 minutes) = 0.9, U(30 minutes) = 0.7, a n d U(40 minutes) = 0.2. (a) Which route will minimize Lynn's expected travel time? (b) Which route will maximize Lynn's utility? (c) When it comes to travel time, is Lynn a risk seeker or a risk avoider?

(a) Broad (b) Expressway (c) Risk avoider

3.20 Mickey Lawson is considering investing some money that he inherited. The following payoff table gives the profits that would be realized during the next year for each of three investment alternatives Mickey is considering: DECISION ALTERNATIVE | GOOD ECONOMY | POOR ECONOMY Stock market 80,000 | -20,000 Bonds 30,000 | 20,000 CDs 23,000 | 23,000 Probability 0.5 | 0.5 (a) What decision would maximize expected profits? (b) What is the maximum amount that should be paid for a perfect forecast of the economy?

(a) Stock market (b) $21,500

3.22 Allen Young has always been proud of his personal investment strategies and has done very well over the past several years. He invests primarily in the stock market. Over the past several months, however, Allen has become very concerned about the stock market as a good investment. In some cases, it would have been better for Allen to have his money in a bank than in the market. During the next year, Allen must decide whether to invest $10,000 in the stock market or in a certificate of deposit (CD) at an interest rate of 9%. If the market is good, Allen believes that he could get a 14% return on his money. With a fair market, he expects to get an 8% return. If the market is bad, he will most likely get no return at all—in other words, the return would be 0%. Allen estimates that the probability of a good market is 0.4, the probability of a fair market is 0.4, and the probability of a bad market is 0.2, and he wishes to maximize his long-run average return. (a) Develop a decision table for this problem. (b) What is the best decision?

(b) CD

During normal business hours on the east coast, calls to the toll-free reservation number of the Nite Time Inn arrive at a rate of 5 per minute. It has been determined that the number of calls per minute can be described by the Poisson distribution. Find the probability that in the next minute, the number of calls arriving will be

. Average number per minute = 5. So λ = 5 (a) P(X is exactly 5) = P(5) = (55e-5)/5! = 0.1755 (b) P(X is exactly 4) = P(4) = (54e-5)/4! = 0.1755 (c) P(X is exactly 3) = P(3) = (53e-5)/3! = 0.1404 (d) P(X is exactly 6) = P(6) = (56e-5)/6! = 0.1462 (e) P(X< 2) = P(X is 0 or 1) = P(0) + P(1) = 0.0067 + 0.0337 = 0.0404

Data for a particular subdivision near Greenville indicate that the average price per square foot for a house is $100 with a standard deviation of $5, following a normal distribution. (a) What is the probability that a potential home buyer will face an average price greater than $90 per square foot for a house? (b) What is the probability that a potential home buyer will face an average price of less than $85 per square foot for a house? (c) What is the probability that a potential home buyer will face an average price of less than $108 per square foot for a house?

. The time to complete the project (X) is normally distributed with m = 40 and s = 5. A penalty must be paid if the project takes longer than the due date (or if X > due date). a) P(X > 40) = 1 - (X £ 40) = 1 - P(Z £ (40 - 40)/5) = 1 - P(Z £ 0) = 1 - 0.5 = 0.5 b) P(X > 43) = 1 - P(X £ 43) = 1 - P(Z £ (43 - 40)/5) = 1 - P(Z £ 0.6) = 1 - 0.72575 = 0.27425 c) If there is a 5% chance that the project will be late, there is a 95% chance the project will be finished by the due date. So, P(X £ due date) = 0.95 or P(X £ _____) = 0.95 The Z-value for a probability of 0.95 is approximately 1.64, so the due date (X) should have a Z-value of 1.64. Thus, <ART FILE="02_42eq01.eps" W="51.25pt" H="19.691pt" XS="100%" YS="100%"/> 5(1.64) = X - 40 X = 48.2. The due date should be 48.2 weeks from the start of the project

2-25 The schedule for the Killers next year is as follows (refer to Problem 2-24): Game 1: The Machos Game 2: Mama's Boys (a) What is the probability that the Killers will win their first game? (b) What is the probability that the Killers will win their last game? (c) What is the probability that the Killers will break even—win exactly one game? (d) What is the probability that the Killers will win every game? (e) What is the probability that the Killers will lose every game? (f) Would you want to be the coach of the Killers?

. a. Probability = P(K over M) = 0.2. b. Probability = P(K over MB) = 0.4. c. Probability = [P(K over M) and P(MB over K)] or [P(K over MB) and P(M over K)] = (0.2)(0.6) + (0.4)(0.8) = 0.12 + 0.32 = 0.44 d. Probability = [P(K over MB) and P(K over M)] = (0.4)(0.2) = 0.08 e. Probability = P(MB over K) and P(M over K) = (0.6)(0.8) = 0.48 f. No. They do not appear to be a very good team.

3.42 Child's Play, a producer of fast-moving consumer goods, is considering producing a new children's soap. The company plans to commission a survey concerning the success or failure of a new product. A market research firm has told the company that there is a 0.65 probability that the survey will be positive given a favorable market. Similarly, there is a 0.40 probability that the survey will be positive given an unfavorable market. These statistics indicate the accuracy of the survey. Prior to contacting the market research firm, the company's best estimate of a favorable market was 50%. Using Bayes' theorem, determine the probability of a favorable market given a favorable survey.

0.62

There are 10 questions on a true-false test. A student feels unprepared for this test and randomly guesses the answer for each of these. (a) What is the probability that the student gets exactly 7 correct? (b) What is the probability that the student gets exactly 8 correct? (c) What is the probability that the student gets exactly 9 correct? (d) What is the probability that the student gets exactly 10 correct? (e) What is the probability that the student gets more than 6 correct?

2 (a) 0.1172 (b) 0.0439 (c) 0.0098 (d) 0.0010 (e) 0.1719

0 Harrington Health Food stocks 5 loaves of NeutroBread. The probability distribution for the sales of Neutro-Bread is listed in the following table. How many loaves will Harrington sell on average? NUMBER OF LOAVES SOLD 0 1 2 3 4 5 PROBABILITY .05 .15 .2 .25 .2 .15

2-30. X P(X) X · P(X) 0 0.05 0.00 1 0.15 0.15 2 0.2 0.40 3 0.25 0.75 4 0.20 0.80 5 0.15 0.75 2.85 Hence, 2.85 loaves will be sold on average.

Megley Cheese Company is a small manufacturer of several different cheese products. One of the products is a cheese spread that is sold to retail outlets. Jason Megley must decide how many cases of cheese spread to manufacture each month. The probability that the demand will be six cases is 0.1, seven cases is 0.3, eight cases is 0.5, and nine cases is 0.1. The cost of every case is $45, and the price that Jason gets for each case is $95. Unfortunately, any cases not sold by the end of the month are of no value, due to spoilage. How many cases of cheese should Jason manufacture each month?

8 cases

BEP = f / (s - n)

An equation to determine the break-even point (BEP) in units as a function of the selling price per unit (s), fixed cost ( f ), and variable cost (n).

Katherine D'Ann, from Problem 1-18, has become concerned that sales may fall, as the team is on a terrible losing streak and attendance has fallen off. In fact, Katherine believes that she will sell only 500 programs for the next game. If it was possible to raise the selling price of the program and still sell 500, what would the price have to be for Katherine to break even by selling 500?

BEP = f/(s - v) 500 = 1400/(s - 3) 500(s - 3) = 1400 s - 3 = 1400/500 s = 2.8 + 3 s = $5.80

3.36 A group of medical professionals is considering the construction of a private clinic. If the medical demand is high (i.e., there is a favorable market for the clinic), the physicians could realize a net profit of $100,000. If the market is not favorable, they could lose $40,000. Of course, they don't have to proceed at all, in which case there is no cost. In the absence of any market data, the physicians' best guess is that there is a 50-50 chance the clinic will be successful. Construct a decision tree to help analyze this problem. What should the medical professionals do?

Construct the clinic (EMV = 30,000).

In a sample of 1,000 representing a survey from the entire population, 650 people were from Laketown, and the rest of the people were from River City. Out of the sample, 19 people had some form of cancer. Thirteen of these people were from Laketown. (a) Are the events of living in Laketown and having some sort of cancer independent? (b) Which city would you prefer to live in, assuming that your main objective was to avoid having cancer?

In the sample of 1,000 people, 650 people were from Laketown and 350 from River City. Thirteen of those with cancer were from Laketown. Six of those with cancer were from River City. a. The probability of a person from Laketown having cancer: P(cancer | laketown) = 13/650 = 1/50 = 0.020 The probability of a person having cancer: P (cancer) = 19/1000 Not independent. b. I would rather live in River City.

Burger City is a large chain of fast-food restaurants specializing in gourmet hamburgers. A mathematical model is now used to predict the success of new restaurants based on location and demographic information for that area. In the past, 70% of all restaurants that were opened were successful. The mathematical model has been tested in the existing restaurants to determine how effective it is. For the restaurants that were successful, 90% of the time the model predicted they would be, while 10% of the time the model predicted a failure. For the restaurants that were not successful, when the mathematical model was applied, 20% of the time it incorrectly predicted a successful restaurant, while 80% of the time it was accurate and predicted an unsuccessful restaurant. If the model is used on a new location and predicts the restaurant will be successful, what is the probability that it actually is successful?

Let S = successful restaurant U = unsuccessful restaurant PS = model predicts successful restaurant PU = model predicts unsuccessful restaurant P(S) = 0.70 P(U) = 0.30 P(PS | S) = 0.90 P(PU | S) = 0.10 P(PS | U) = 0.20 P(PU | U) = 0.80 .91

Policy Pollsters is a market research firm specializing in political polls. Records indicate in past elections, when a candidate was elected, Policy Pollsters had accurately predicted this 80% of the time and was wrong 20% of the time. Records also show, for losing candidates, Policy Pollsters accurately predicted they would lose 90% of the time and was wrong only 10% of the time. Before the poll is taken, there is a 50% chance of winning the election. If Policy Pollsters predicts a candidate will win the election, what is the probability that the candidate will actually win? If Policy Pollsters predicts that a candidate will lose the election, what is the probability that the candidate will actually lose?

Let W = candidate wins the election L = candidate loses the election PW = poll predicts win PL = poll predicts loss P(W) = 0.50 P(L) = 0.50 P(PW | W) = 0.80 P(PL | W) = 0.20 P(PW | L) = 0.10 P(PL | L) = 0.90

Although Ken Brown (discussed in Problem 3-17) is the principal owner of Brown Oil, his brother Bob is credited with making the company a financial success. Bob is vice president of finance. Bob attributes his success to his pessimistic attitude about business and the oil industry. Given the information from Problem 3-17, it is likely that Bob will arrive at a different decision. What decision criterion should Bob use, and what alternative will he select?

Maximin criterion; Texan

Compute the probability of "loaded die given that a 3 was rolled," as shown in the example in Section 2.3, this time using the general form of Bayes' Theorem from Equation 2-5.

P(AIB) = P(B|A)P(A) / P(B|A)P(A) + P(B|A)P(A) where A = fair die = F = unfair die = L B = getting a 3 P(AIB) = P(F | 3) = P(3 I F)P(F) / P(3 | F)P(F)+P(3 | L)P(L) (.166)(.5)/(.166)(.5)+ (.6)(.5) .083 / .083 + .03 = 0.22 Therefore, P(L) = 1 - 0.22 = 0.78

Which of the following are probability distributions? Why? (a) RANDOM VARIABLE X PROBABILITY 2 0.1 -1 0.2 0 0.3 1 0.25 2 0.15 (b) RANDOM VARIABLE Y PROBABILITY 1 1.1 1.5 0.2 2 0.3 2.5 0.25 3 -1.25 RANDOM VARIABLE Z PROBABILITY 1 0.1 2 0.2 3 0.3 4 0.4 5 0.0

Parts (a) and (c) are probability distributions because the probability values for each event are between 0 and 1, and the sum of the probability values for the events is 1. 2-30. X P(X) X · P(X) 0 0.05 0.00 1 0.15 0.15 2 0.2 0.40 3 0.25 0.75 4 0.20 0.80 5 0.15 0.75 2.85 Hence, 2.85 loaves will be sold on average. 2-31. X P(X) X · P(X) X - E(X) (X - E(X))2 (X - E(X))2P(X) 1 0.05 0.05 -4.45 19.803 0.99013 2 0.05 0.1 -3.45 11.903 0.59513 3 0.10 0.3 -2.45 6.003 0.60025 4 0.10 0.4 -1.45 2.103 0.21025 5 0.15 0.75 -0.45 0.203 0.03038 6 0.15 0.9 0.55 0.303 0.04538 7 0.25 1.75 1.55 2.403 0.60063 8 0.15 1.20 2.55 6.5025 0.97538 5.45 4.04753 E(X) = 5.45 s2 = S[X - E(X)]2P(X) = 4.0475

The Northside rifle team has two markspersons, Dick and Sally. Dick hits a bull's-eye 90% of the time, and Sally hits a bull's-eye 95% of the time. (a) What is the probability that either Dick or Sally or both will hit the bull's-eye if each takes one shot? (b) What is the probability that Dick and Sally will both hit the bull's-eye? (c) Did you make any assumptions in answering the preceding questions? If you answered yes, do you think that you are justified in making the assumption(s)?

The probability of Dick hitting the bull's-eye: P(D) = 0.90 The probability of Sally hitting the bull's-eye: P(S) = 0.95 a. The probability of either Dick or Sally hitting the bull's-eye: P(D or S) = P(D) + P(S) - P(D)P(S) = 0.90 + 0.95 - (0.90)(0.95) = 0.995 b. P(D and S) = P(D)P(S) = (0.9)(0.95) = 0.855 c. It was assumed that the events are independent. This assumption seems to be justified. Dick's performance shouldn't influence Sally's performance.

The total fixed cost per game includes salaries, rental fees, and cost of the workers in the six booths. These are: Salaries = $20,000 Rental fees = 2,400 = $2 = $4,800 Booth worker wages = 6 = 6 = 5 = $7 = $1,260 Total fixed cost per game = $20,000 = $4,800 = $1,260 = $26,060 The cost of this allocated to each food item is shown in the table: https://i.gyazo.com/6292e59d2987e3556ae772ee1739d3d9.png

To determine the total sales for each item that is required to break even, multiply the selling price by the break even volume. The results are shown: Soft drink $1.50 8686.67 $13,030.00 Coffee $2.00 4343.33 $8,686.67 Hamburgers $2.50 3474.67 $8,686.67 Hot dogs $2.00 4343.33 $8,686.67 Misc. snacks $1.00 4343.33 $4,343.33 to break even, the total sales must be $43,433.34.

5 Zoe Garcia is the manager of a small office-support business that supplies copying, binding, and other services for local companies. Zoe must replace a worn-out copy machine that is used for black-andwhite copying. Two machines are being considered, and each of these has a monthly lease cost plus a cost for each page that is copied. Machine 1 has a monthly lease cost of $600, and there is a cost of $0.010 per page copied. Machine 2 has a monthly lease cost of $400, and there is a cost of $0.015 per page copied. Customers are charged $0.05 per page for copies. (a) What is the break-even point for each machine? (b) If Zoe expects to make 10,000 copies per month, what would be the cost for each machine? (c) If Zoe expects to make 30,000 copies per month, what would be the cost for each machine? (d) At what volume (the number of copies) would the two machines have the same monthly cost? What would the total revenue be for this number of copies?

a) Machine 1: f = 600 s = 0.05 v = 0.010 BEP = f/(s - v) = 600/(0.05 - 0.010) = 15,000 Machine 2: f = 400 s = 0.05 v = 0.015 BEP = f/(s - v) = 400/(0.05 - 0.015) = 11,428.57 b) Machine 1: Cost = 600 + 0.010(10,000) = $700 Machine 2: Cost = 400 + 0.015(10,000) = $550 c) Machine 1: Cost = 600 + 0.010(30,000) = $900 Machine 2: Cost = 400 + 0.015(30,000) = $850 d) Let X = the number of copies 600 + 0.010X = 400 + 0.015X 600 - 400 = 0.015X - 0.010X 200 = 0.05X X = 40,000 copies

A couple of entrepreneurial business students at State University decided to put their education into practice by developing a tutoring company for business students. While private tutoring was offered, it was determined that group tutoring before tests in the large statistics classes would be most beneficial. The students rented a room close to campus for $300 for 3 hours. They developed handouts based on past tests, and these handouts (including color graphs) cost $5 each. The tutor was paid $25 per hour, for a total of $75 for each tutoring session. (a) If students are charged $20 to attend the session, how many students must enroll for the company to break even? (b) A somewhat smaller room is available for $200 for 3 hours. The company is considering this possibility. How would this affect the break-even point?

a) f = 300 + 75 = 375 s = 20 v = 5 BEP = f/(s - v) = 375/(20 - 5) = 25 b) f = 200 + 75 = 275 s = 20 v = 5 BEP = f/(s - v) = 275/(20 - 5) = 18.333

Golden Age Retirement Planners specializes in providing financial advice for people planning for a comfortable retirement. The company offers seminars on the important topic of retirement planning. For a typical seminar, the room rental at a hotel is $1,000, and the cost of advertising and other incidentals is about $10,000 per seminar. The cost of the materials and special gifts for each attendee is $60 per person attending the seminar. The company charges $250 per person to attend the seminar, as this seems to be competitive with other companies in the same business. How many people must attend each seminar for Golden Age to break even?

f = 11000 s = 250 v = 60 BEP = f/(s - v) = 11000/(250 - 60) = 57.9

1-17 Ray Bond, from Problem 1-16, is trying to find a new supplier that will reduce his variable cost of production to $15 per unit. If he was able to succeed in reducing this cost, what would the break-even point be?

f = 150 s = 50 v = 15 BEP = f/(s - v) = 150/(50 - 15) = 4.29 units

1-16 Ray Bond sells handcrafted yard decorations at county fairs. The variable cost to make these is $20 each, and he sells them for $50. The cost to rent a booth at the fair is $150. How many of these must Ray sell to break even?

f = 150 s = 50 v = 20 BEP = f/(s - v) = 150/(50 - 20) = 5 units

Farris Billiard Supply sells all types of billiard equipment and is considering manufacturing its own brand of pool cues. Mysti Farris, the production manager, is currently investigating the production of a standard house pool cue that should be very popular. Upon analyzing the costs, Mysti determines that the materials and labor cost for each cue is $25 and the fixed cost that must be covered is $2,400 per week. With a selling price of $40 each, how many pool cues must be sold to break even? What would the total revenue be at this break-even point?

f = 2400 s = 40 v = 25 BEP = f/(s - v) = 2400/(40 - 25) = 160 per week Total revenue = 40(160) = $6400

Mysti Farris (see Problem 1-20) is considering raising the selling price of each cue to $50 instead of $40. If this is done while the costs remain the same, what would the new break-even point be? What would the total revenue be at this break-even point?

f = 2400 s = 50 v = 25 BEP = f/(s - v) = 2400/(50 - 25) = 96 per week Total revenue = 50(96) = $4800

Mysti Farris (see Problem 1-20) believes that there is a high probability that 120 pool cues can be sold if the selling price is appropriately set. What selling price would cause the break-even point to be 120?

f = 2400 s = ? v = 25 BEP = f/(s - v) 120 = 2400/(s - 25) 120(s - 25) = 2400 s = 45

Gina Fox has started her own company, Foxy Shirts, which manufactures imprinted shirts for special occasions. Since she has just begun this operation, she rents the equipment from a local printing shop when necessary. The cost of using the equipment is $350. The materials used in one shirt cost $8, and Gina can sell these for $15 each. (a) If Gina sells 20 shirts, what will her total revenue be? What will her total variable cost be? (b) How many shirts must Gina sell to break even? What is the total revenue for this?

f = 350 s = 15 v = 8 a) Total revenue = 20(15) = $300 Total variable cost = 20(8) = $160 b) BEP = f/(s - v) = 350/(15 - 8) = 50 units Total revenue = 50(15) = $750

1-18 Katherine D'Ann is planning to finance her college education by selling programs at the football games for State University. There is a fixed cost of $400 for printing these programs, and the variable cost is $3. There is also a $1,000 fee that is paid to the university for the right to sell these programs. If Katherine was able to sell programs for $5 each, how many would she have to sell in order to break even?

f = 400 + 1,000 = 1,400 s = 5 v = 3 BEP = f/(s - v) = 1400/(5 - 3) = 700 units

Key Equation

https://i.gyazo.com/8581bfb3dc4e21bd280d34f2617f1a4f.png https://i.gyazo.com/24aa3f42af930fabcb06114600c66d79.png https://i.gyazo.com/649390360ae746dab72f735af29b3019.png https://i.gyazo.com/4622b8613a1dea396545f7867f7296ef.png

7 An industrial oven used to cure sand cores for a factory manufacturing engine blocks for small cars is able to maintain fairly constant temperatures. The temperature range of the oven follows a normal distribution with a mean of 450°F and a standard deviation of 25°F. Leslie Larsen, president of the factory, is concerned about the large number of defective cores that have been produced in the past several months. If the oven gets hotter than 475°F, the core is defective. What is the probability that the oven will cause a core to be defective? What is the probability that the temperature of the oven will range from 460°F to 470°F?

m = 450 degrees s = 25 degrees X = 475 degrees Z = X - U / O = 475 - 450 / 25 = 1 The area between X1 and X2 is 0.78814 - 0.65542 = 0.13272. Thus, the probability of being between 460 and 470 degrees is = 0.1327.

Armstrong Faber produces a standard number-two pencil called Ultra-Lite. Since Chuck Armstrong started Armstrong Faber, sales have grown steadily. With the increase in the price of wood products, however, Chuck has been forced to increase the price of the Ultra-Lite pencils. As a result, the demand for Ultra-Lite has been fairly stable over the past 6 years. On average, Armstrong Faber has sold 457,000 pencils each year. Furthermore, 90% of the time sales have been between 454,000 and 460,000 pencils. It is expected that the sales follow a normal distribution with a mean of 457,000 pencils. Estimate the standard deviation of this distribution. (Hint: Work backward from the normal table to find Z. Then apply Equation 2-13.)

m = 457,000 Ninety percent of the time, sales have been between 460,000 and 454,000 pencils. This means that 10% of the time sales have exceeded 460,000 or fallen below 454,000. Since the curve is symmetrical, we assume that 5% of the area lies to the right of 460,000 and 5% lies to the left of 454,000. Thus, 95% of the area under the curve lies to the left of 460,000. From Table 2.9, we note that the number nearest 0.9500 is 0.94950. This corresponds to a Z value of 1.64. Therefore, we may conclude that the Z value corresponding to a sale of 460,000 pencils is 1.64. Using Equation 2-13, we get <ART FILE="02_40eq01.eps" W="47.999pt" H="25pt" XS="100%" YS="100%"/> X = 460,000 m = 457,000 s is unknown Z = 1.64 <ART FILE="02_40eq02.eps" W="98.361pt" H="21.105pt" XS="100%" YS="100%"/> 1.64 s = 3000 <ART FILE="02_40eq03.eps" W="19.855pt" H="19.25pt" XS="100%" YS="100%"/> = 1829.27

Profit = sX - f - vX

s = selling price per unit f = fixed cost v = variable cost per unit X = number of units sold An equation to determine profit as a function of the selling price per unit, fixed cost, variable cost, and number of units sold.