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Statistics For Business Decision Making And Analysis 2nd Edition Robert Stine, Dean Foster - Solutions

1. Independent events 2. Disjoint events 3. Union 4. Intersection 5. Complement of A 6. Sample space 7. Addition Rule 8. Complement Rule 9. Boole’s inequality 10. Dependent events (a) P (A and B) = P (A) × P(B) (b) Ac (c) S (d) P (A or B) + P (A and B) = P (A) + P(B) (e) P (A or B) . P (A) + P
1. The sample space S for this experiment has 10 elements. 2. The assumption of independence implies that each shopper has the same probability for carrying a bag. 3. P (A) + P (B) = P (A or B). 4. The probability that both events B and C occur is equal to P (B). 5. The probability that a randomly
1. The events A = {3 candidates on Monday rate above 7} and B = {two candidates on Tuesday rate above 7} are disjoint events. 2. If 8 of the 12 candidates on Monday and Tuesday rate above 6, then the probability of a candidate rating above 6 on Wednesday is 8>12. 3. The HR group has monitored the
Each of the following scatterplots shows a sequence of observations; the x-axis enumerates the sequence as in the example of the arrival of calls to the agent at the help desk in this chapter. For which cases does the Law of Large Numbers apply to probabilities based on these observations?
As in Exercise 23, these scatterplots graph a sequence of observations taken under similar conditions. If we could watch these processes longer and longer and accumulate more data, in which cases would the Law of Large Numbers apply?
A shopper in a convenience store can make a food selection from frozen items, refrigerated packages, fresh foods, or deli items. Let the event A = {frozen, refrigerated, fresh} and B = {fresh, deli}. (a) Find the intersection A and B. (b) Find the union of A and B. (c) Find the event Ac.
A credit-rating agency assigns ratings to corporate bonds. The agency rates bonds offered to companies that are most likely to honor their liabilities AAA. The ratings fall as the company becomes more likely to default, dropping from AAA to AA, A, down to BBB, BB, B, CCC, CC, R, and then D (for in
A brand of men’s pants offered for sale at a clothing store comes in various sizes. The possible waist sizes are Waist: {24 inches, 26 inches, c, 46 inches}with inseams (length of the pant leg) Inseam: {28 inches, 29 inches, c, 40 inches} Define the event B = {waist 40 inches or larger} and T =
An auto dealer sells several brands of domestic and foreign cars in several price ranges. The brands sold by the dealer are Saturn, Pontiac, Buick, Acura, and Ferrari. The prices of cars range from under $20,000 to well over $100,000 for a luxury car. Define the event D = {Saturn, Pontiac, Buick}
Which of the following implications comes from the Law of Large Numbers? (a) Independence produces a sequence of observations without patterns. (b) Probability calculations require a large number of experiments. (c) Proportions get close to probabilities in the long run.
Count the number of cars that pass by an intersection during consecutive five-minute periods on a highway leading into a city. Would these data allow us to use the Law of Large Numbers eventually to learn P(more than 50 cars in five minutes)?
An airline would like to know the probability of a piece of passenger’s luggage weighing more than 40 pounds. To learn this probability, a baggage handler picks off every 20th bag from the conveyor and weighs it. Do you think these data allow the airline to use the Law of Large Numbers eventually
A Web site recorded whether visitors click on a shown and. The following plot shows the outcomes for a sequence of 100 visitors, with a 1 shown if the visitor clicked on the ad and a 0 otherwise.(a) Does it appear that the Law of Large Numbers is applicable if this sequence continues
A basketball team is down by 2 points with only a few seconds remaining in the game. There’s a 50% chance that the team will be able to make a 2-point shot and tie the game, compared to a 30% chance that it will make a 3-point shot and win. If the game ends in a tie, the game continues to
It’s time for an advertising firm to renew its contract with a client. The advertising firm wants the client to increase the amount it spends to advertise. If the firm proposes to continue the current contract, there’s a 75% chance that the client will accept the proposal. To increase the
In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are wary of flying because they just learned of an accident. (a) A travel agent suggests that, since the Law of Large Numbers makes it highly unlikely to have two plane crashes within a
Many construction sites post a sign that proclaims the number of days since the last accident that injured an employee. (a) A friend who walks by the construction site tells you to expect to see an ambulance any day. The count has grown from 20 days to 60 days to 100 days. An accident is coming any
The Web site for M&M™ candies claimed that 24% of plain M&M candies are blue, 20% are orange, 16% green, 14% yellow, and 13% each red and brown. (a) Pick one M&M at random from a package. 1. Describe the sample space. 2. What is the probability that the one you pick is blue or red? 3. What is
At a large assembly line, the manufacturer reassigns employees to different tasks each month. This keeps the workers from getting bored by endlessly repeating the same task. It also lets them see how the work done in different stages must fit together. It is known that 55% of the employees have
A survey found that 62% of callers in the United States complain about the service they receive from a call center if they suspect that the agent who handled the call is foreign.11 Given this context, what is the probability that (state your assumptions)(a) The next three consecutive callers
A recently installed assembly line has problems with intermittent breakdowns. Recently, it seems that the equipment fails at some point during 15% of the eight-hour shifts that the plant operates. Each day contains three consecutive shifts. State your assumptions. What is the probability that the
Modern manufacturing techniques rely on highly reliable components. Because of the difficulty and time to test the separate components of a computer system, for example, these are often completely assembled before the power is ever applied. Assume that the computer fails to work properly if any
A chain of fast-food restaurants offers an instant-winner promotion. Drink cups and packages of French fries come with peel-off coupons. According to the promotion, 1 in 20 packages is a winner.(a) A group of friends buys four drinks and three orders of fries. What’s the probability that they
A friend of yours took a multiple-choice quiz and got all six questions right, but now claims to have guessed on every question. If each question offers four possible answers, do you believe her? Explain why you either believe her or not.
A down side to running a call center is the high turnover in the operators who answer calls. As an alternative, some companies have turned to €œhomeshoring.€ Rather than have operators come to a call center, the company uses technology to route calls to the operators at home. Recent analysis
Intel and AMD compete to put their processing chips into personal computers. In the third quarter of 2011, Intel claimed 82% of the laptop market and 76% of the market for desktop computers. In 2010, Intel commanded 86% of the laptop market and 72% of the desktop market.(a) What is the probability
The social networking site Facebook is popular in the United States and around the world. This table shows the number of Facebook users and populations in four countries in 2011. All of these counts are in mil-lions of people.13(a) If we select a Facebook member from these at random, what is the
A study reported in The New England Journal of Medicine revealed surprisingly large differences in rates of lung cancer among smokers of different races.14 For each group, the study reported a rate among smokers per 100,000.(a) What is the probability that a black male smoker develops lung
To boost interest in its big sale, a retailer offers special scratch-off coupons. When the shopper makes a purchase, the clerk scratches off the covering to reveal the amount of the discount. The discount is 10%, 20%, 30%, 40%, or 50% of the initial amount. Most of the coupons are for 10% off, with
A fast-food chain randomly attaches coupons for prizes to the packages used to serve french fries. Most of the coupons say “Play again,” but a few are winners. Seventy-five percent of the coupons pay nothing, with the rest evenly divided between “Win a free order of fries” and “Win a free
In an NBA basketball game on January 22, 2006, Kobe Bryant of the Los Angeles Lakers scored 81 points, a total score second only to a 100-point performance by Wilt Chamberlain in 1962. The data file [kobe] contains the sequence of his attempts, with 0 indicating a miss, 2 for a made regular basket,
Racetracks give the odds for each horse winning a race rather than the probability. Here are the odds for the 2011 Belmont Stakes.Odds at a racetrack indicate the amount won for a bet on the winning horse. For example, these racetrack odds mean that a $1 bet on Master of Hounds wins a $5 payout
Many businesses use auditors to check over their operations. With thousands of transactions, it can be easy for a crook to slip in a few bogus checks. Sales fraud occurs when managers whose pay is based on performance inflate reported sales figures in order to get a bonus or higher salary. One way
1. Probability of B given A 2. Probability of Bc given A 3. Identifies independent events 4. Identifies dependent events 5. Bayes’ Rule 6. Multiplication Rule (a) P (A and B) = P (A) × P (B∣ A) (b) P (A∣ B) = P (B∣A) × P (A) / P (B) (c) P (A) = P (A∣B) (d) 1 – P (B∣A) (e) P(A)
1. If A1 is independent of A2, P (A) | A2) = P (A2). Exercises 15–20. The Human Resources division classifies employees of the firm into one of three categories: administrative, clerical, or management. Suppose we choose an employee at random. Define the events A = {administrative}, C =
(a) Many companies test employees for recreational drug use. What simple diagnostic test procedure for whether the employee has used drugs has sensitivity equal to 1? Recall that the sensitivity of a test procedure is the probability that the test detects the indicated problem given that the
An insurer is studying the characteristics of those who buy its policies. It discovered that, among young drivers, 45% insure a foreign-made car. Among those who drive foreign-made cars, the insurer also discovered that 45% are young. Consider the events.Y = {randomly chosen driver is young}F =
Does the insurer described in Exercise 31 cover more drivers who are young or more drivers who insure foreign-made cars?
A recent study looked into the amount of debt accumulated by recent college graduates. The study found that, among those with student loans, 42% said working during college affected their grades?13 (a) Convert this statement into a conditional probability, including a short description of the
The study of recent college graduates described in Exercise 33 found that among those who had graduated with debts from student loans, 33% had sold possessions since graduating. Among those who had graduated free of debt, only 17% had sold possessions since graduating. (a) Express these two
An auto manufacturer has studied the choice of options available to customers when ordering a new vehicle. Of the customers who order a car, 25% choose a sunroof, 35% choose a leather interior, and 10% choose both. (The rest opt for neither.) Of the customers who order a truck, 20% choose an
A vendor specializing in outdoor gear offers customers three ways to place orders for home delivery. Customers can place orders either in the store, online, or via telephone using a catalog. Of the orders, one-half are made online with one-quarter each made in stores or via telephone. Of the online
Seventy percent of service calls regarding kitchen appliances involve a minor repair. For example, the customer did not read the instructions correctly or perhaps failed to connect the appliance correctly. These service calls cost +75. Service calls for major repairs are more expensive. More than
Some electronic devices are better used than new: The failure rate is higher when they are new than when they are six months old. For example, half of the personal music players of a particular brand have a flaw. If the player has the flaw, it dies in the first six months. If it does not have this
Some of the managers at a company have an MBA degree. Of managers at the level of director or higher, 60% have an MBA. Among other managers of lower rank, 35% have an MBA. For this company, 15% of managers have a position at the level of director or higher. If you meet an MBA from this firm, what
A shipment of assembly parts from a vendor offering inexpensive parts is used in a manufacturing plant. The box of 12 parts contains 5 that are defective and will not ft during assembly. A worker picks parts one at a time and attempts to install them. Find the probability of each outcome. (a) The
Many retail stores open their doors early the day after Thanksgiving to attract shoppers to special bargains. A shopper wanting two medium blouses heads for the sale rack, which is a mess, with sizes jumbled together. Hanging on the rack are 4 medium, 10 large, and 6 extra-large blouses. Find the
After assembling an order for 12 computer systems, the assembler noticed that an electronic component that was to have been installed was left over. The assembler then checked the 12 systems in order to find the system missing the component. Assume that he checks them in a random order. (a) What is
Among employees at a retailer, there is a 15% chance that an employee will be absent from work for some reason. The floor manager has been talking with two randomly selected employees each day to get suggestions to improve sales. (a) What is the probability that the first employee he chooses to
Choice leads for developing new business are randomly assigned to 50 employees who make up the direct sales team. Half of the sales team is male, and half is female. An employee can receive at most one choice lead per day. On a particular day, five choice leads are assigned.(a) Are the events
Modern assembly relies on parts built to specifications (spec) so that the components ft together to make the whole. An appliance requires two components, Type A and Type B, supplied by different manufacturers. A robot randomly selects a component from a batch of Type A components and a second
The following table gives the percentages of men and women 20 years and older employed in various occupations in the U.S. workforce in 2010, from the Current Population Survey. Each column sums to 100%. There are about 70 million men and 70 million women in the workforce.(a) The number 14 in the
A company buys components from two suppliers. One produces components that are of higher quality than the other. The high-quality supplier, call it Supplier A, has a defect rate of 2%. The low-quality supplier, Supplier B, has a defect rate of 10% but offers lower prices. This company buys in equal
You fly from Philadelphia to San Francisco with a connection in Dallas. The probability that your fight from Philadelphia to Dallas arrives on time is 0.8. If you arrive on time, then the probability that your luggage makes the connection to San Francisco is 0.9. If you are delayed, then the chance
A survey reports that 62% of callers to a help desk complain about the service if they think they spoke to a foreign agent, compared to 31% who complain if they think they spoke to a native agent. Suppose that 40% of all calls to service centers are handled by agents in the United States. If a
Recent surveys report that although Internet access has grown rapidly, it’s not universal: Only 71% of U.S. households have Internet access at home. Internet access isn’t universal for households that have computers. Of the 77% of households that have computers, 8% do not have Internet
Suppose that 10% of the clerical staff in an office smoke cigarettes. Research shows that 60% of smokers and 15% of nonsmokers suffer a breathing illness by age 65. (a) Do these percentages indicate that smoking and this breathing illness are independent? (b) What’s the probability that a
Many supermarkets offer customers discounts in return for using a shopper's card. These stores also have scanner equipment that records the items purchased by the customer. The combination of a shopping card and scanner software allow the supermarket to track which items customers regularly
Credit-card fraud costs business in the United States billions of dollars each year in stolen goods. Compounding the problem, the risk of fraud increases with the rapidly growing online retail market. To reduce fraud, businesses and credit card vendors have devised systems that recognize
1. The probability of an employee being absent is greater than the probability that the employee is absent given that the employee is sick. 2. The probability that an employee is sick when it is known that the employee is absent is equal to the probability that an employee is absent when it is
1. Expected value of X 2. Variance of X 3. Standard deviation of X 4. Shorthand notation for P(X = x) 5. Has 10 times the standard deviation of X 6. Is always equal to zero 7. Increases the mean of X by 10 8. Has standard deviation 1 (a) E(X - μ) (b) 10X (c) (X - 0.04) > σ (d) X + 10 (e) E(X -
1. If the mean of Y is $3,200, then the variance of Y must be larger than $3,200. 2. If E(X) = +2,300, then P(X . 2,300) = 1 / 2. 3. The units of both the mean and standard deviation of Y are dollars. 4. If an insurance policy limits the coverage of an accident to $500,000, then both E(X) and E(Y)
Which is larger, P (W > 0) or P(Y > 0)?
The lower quartile of a random variable X is the smallest q so that P(X ‰¤ q) ‰¥ 0.25. (This is analogous to the lower edge of the box in the boxplot.) What€™s the lower quartile of X?
A game involving chance is said to be a fair game if the expected amount won or lost is zero. Consider the following arcade game. A player pays $1 and chooses a number from 1 to 10. A spinning wheel then randomly selects a number from 1 to 10. If the numbers match, the player wins $5. Otherwise the
Many state governments use lotteries to raise money for public programs. In a common type of lottery, a customer buys a ticket with a three-digit number from 000 to 999. A machine (such as one with bouncing balls numbered 0 to 9) then selects a number in this range at random. Each ticket bought by
Another way to define a fair game is that a player’s probability of winning must be equal to the player’s share of the pot of money awarded to the winner. All money is put into a pot at the start of the game, and the winner claims the entire amount that is in the pot. (Compare this definition
Two companies are competing to define a new standard for digital recording. Only one standard will be adopted commercially, and it is unclear which standard the consumer market will choose. Company A invests $10 million to promote its standard, and Company B invests $20 million to promote its. If
Given that the random variable X has mean μ = 120 and SD σ = 15, find the mean and SD of each of these random variables that are defined from X: (a) X / 3 (b) 2X – 100 (c) X + 2 (d) X – X
Given that the random variable Y has mean μ = 25 and SD σ = 10, fnd the mean and SD of each of these random variables that are defined from Y: (a) 2Y + 20 (b) 3Y (c) Y / 2 + 0.5 (d) 6 – Y
An investor buys the stock of two companies, investing $10,000 in each. The stock of each company either goes up by 80% after a month (rising to $18,000) with probability 1>2 or drops by 60% (falling to $4,000) with probability 1>2. Assume that the changes in each are independent. Let the
Imagine that the investor in Exercise 37 invests $10,000 for one month in a company whose stock either goes up by 80% after a month with probability 1>2 or drops 60% with probability 1 > 2. After one month, the investor sells this stock and uses the proceeds to buy stock in a second company.
A construction firm bids on a contract. It anticipates a profit of +50,000 if it gets the contract for the full project, and it estimates its profit to be +20,000 on a shared project. The firm estimates there’s a 20% chance it will get the full contract and a 75% chance it will get the shared
A law firm takes cases on a contingent fee basis. If the case goes to trial, the firm expects to earn $25,000 as part of the settlement if it wins and nothing if it does not. The firm wins one-third of the cases that go trial. If the case does not go to trial, the firm earns nothing. Half of the
Even with email, offices use a lot of paper. The manager of an office has estimated how many reams (packs of 500 sheets) are used each day and has assigned these probabilities.(a) How many reams should the manager expect to use each day?(b) What is the standard deviation of the number of reams?(c)
The maintenance staff of a large office building regularly replaces fluorescent ceiling lights that have gone out. During a visit to a typical floor, the staff may have to replace several lights. The manager of this staff has given the following probabilities to the number of lights (identifed by
Companies based in the United States that operate in foreign nations must convert currencies in order to pay obligations denoted in those currencies. Consider a company that has a contract that requires it to pay 1,000,000 Venezuelan bolivars for an oil shipment in six months. Because of economic
During the 1960s and into the 1970s, the Mexican government pegged the value of the Mexican peso to the U.S. dollar at 12 pesos per dollar. Because interest rates in Mexico were higher than those in the United States, many investors (including banks) bought bonds in Mexico to earn higher returns
A manufacturer of inexpensive printers offers a model that retails for $150. Each sale of one of these models earns the manufacturer $60 in profits. The manufacturer is considering offering a $30 mail-in rebate. Assume that a randomly selected customer has probability p of purchasing this model of
With homeowners buying valuable appliances such as plasma TVs and electronic appliances, there’s more concern about the cost of being hit by lightning.11 Suppose that you’ve just bought a $4,000 fat-screen TV. Should you also buy the $50 surge protector that guarantees to protect your TV from
An insurance salesman visits up to three clients each day, hoping to sell a new policy. He stops for the day once he makes a sale. Each client independently decides whether to buy a policy; 10% of clients purchase the policy. (a) Create a probability model for the number of clients the salesman
An office machine costs $10,000 to replace unless a mysterious, hard-to-find problem can be found and fixed. Repair calls from any service technician cost $500 each, and you’re willing to spend up to $2,000 to get this machine fixed. You estimate that a repair technician has a 25% chance of
The ATM at a local convenience store allows customers to make withdrawals of $10, $20, $50, or $100. Let X denote a random variable that indicates the amount withdrawn by a customer. The probability distribution of X isP (10) = 0.2P (20) = 0.5P (50) = 0.2P (100) = 0.1(a) Draw the probability
A company orders components from Japan for its game player. The prices for the items that it orders are in Japanese yen. When the products are delivered, it must convert dollars into yen to pay the Japanese producer. When its next order is delivered (and must be paid for), it believes that the
Star basketball player Derrick Rose of the Chicago Bulls was named Most Valuable Player for the 2010-2011 NBA season.12 This table lists the number of baskets made of several types along with the number of attempts.Use the implied percentages for each type of shot as probabilities to answer the
In 13 professional football seasons from 1999 to 2011, kicker David Akers recorded the following results for field goal attempts of various distances.Use the implied percentages as probabilities to answer the following questions: (a) A made field goal scores 3 points. What is the expected value (in
These data are daily percentage changes of several stocks in 2010 and 2011. This question focuses on stock in Apple. (a) Describe the shape and any key features of the histogram of percentage changes in Apple stock. (b) Find the mean and standard deviation of the percentage changes in Apple stock.
This exercise considers daily percentage changes in stock of Disney during 2010 and 2011. Use the same data as in Exercise 53. (a) Describe the shape and any key features of the histogram. (b) Find the mean and standard deviation of the percentage changes. Use the properties of these data to define
A construction company manages two large building projects. Both require about 50 employees with comparable skills. At a meeting, the site managers from the two projects got together to estimate the labor needs of their projects during the coming winter. At the end of the meeting, each manager
Credit scores rate the quality of a borrower, suggesting the chances that a borrower will repay a loan. Borrowers with higher scores are more likely to repay loans on time. Borrowers with lower scores are more likely to default. For example, those with scores below 600 have missed 2.5 payments on
1. The function defined as p(1) = 1 / 2, p(2) = 1 / 4, p(3) = 1 / 8, p(4) = 1 / 8, and p(x) = 0 otherwise is a probability distribution that could be used to model X.2. The function defined as p(1) = 1 / 2, p(2) = 1 / 4, p(3) = 1 / 2, p(4) = -1 / 4, and p(x) = 0 otherwise is a probability
1. Consequence of positive covariance 2. Covariance between X and Y. 3. Property of uncorrelated random variables 4. Weighted sum of two random variables 5. Sharpe ratio of a random variable 6. Implies X and Y are independent random variables 7. Implies X and Y are identically distributed 8. Symbol
1. If managers believe that costs for both fuels tend to rise and fall together, then they should model X and Y as independent. 2. A negative covariance between X and Y would increase the uncertainty about future costs. 3. Because the means and SDs of these random variables are the same, the random
1. On average, retailers expect to sell the same amount on the first weekend and the second weekend. 2. If retail sales are exceptionally low on the first weekend, retailers expect lower than average sales on the following weekend. 3. The standard deviation of total sales over two consecutive
What is the covariance between the random variable Y and itself, Cov1Y, Y2? What is the correlation between a random variable and itself?
If an investor decides on the all-Microsoft portfolio (in the text example), does it make sense to use the Sharpe ratio to decide whether to invest $2,000 or $4,000?
Kitchen remodeling is a popular way to improve the value of a home. (a) If X denotes the amount spent for labor and Y the cost for new appliances, do you think these would be positively correlated, negatively correlated, or independent? (b) If a family is on a strict budget that limits the amount
An insurance company has studied the costs from covering auto accidents. Its policies offer separate coverage for personal injury and damage to property. It has found that the amount paid in these two categories is highly positively correlated. What is the impact of this dependence on the total
Independent random variables X and Y have the means and standard deviations as given in the following table. Use these parameters to find the expected value and SD of the following random variables that are derived from X and Y:(a) 2X - 100(b) 0.5Y(c) X + Y(d) X €“ Y
The independent random variables X and Y have the means and standard deviations as given in the following table. Use these parameters to find the expected value and SD of the following random variables that are derived from X and Y:(a) 8X(b) 3Y - 2(c) 1X + Y2 / 2(d) X - 2Y

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