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probability statistics
The Practice Of Statistics For Business And Economics 3rd Edition David S. Moore, George P. McCabe, Layth C. Alwan, Bruce A. Craig, William M. Duckworth - Solutions
Independent? In the setting of Exercise 5.17, are events A and B independent? Do a calculation that proves your answer.
Tastes in music. Musical styles other than rock and pop are becoming more popular. A survey of college students finds that 40% like country music, 30% like gospel music, and 10%like both.(a) Make a Venn diagram with these results.(b) What percent of college students like country but not gospel?(c)
Will we get the jobs? Consolidated Builders has bid on two large construction projects. The company president believes that the probability of winning the first contract (event A)is 0.6, that the probability of winning the second (event B) is 0.5, and that the probability of winning both jobs
Getting into an MBA program. Ramon has applied to MBA programs at both Harvard and Stanford. He thinks the probability that Harvard will admit him is 0.4, the probability that Stanford will admit him is 0.5, and the probability that both will admit him is 0.2.(a) Make a Venn diagram with the
A random walk onWall Street? The “random walk” theory of securities prices holds that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in any
Nonconforming chips. Automobiles use semiconductor chips for engine and emission control, repair diagnosis, and other purposes. An auto manufacturer buys chips from a supplier. The supplier sends a shipment of which 5% fail to conform to performance specifications. Each chip chosen from this
Playing the lottery. An instant lottery game gives you probability 0.02 of winning on any one play. Plays are independent of each other. If you play 5 times, what is the probability that you win at least once?
Hiring strategy. A chief executive officer (CEO) has resources to hire one vice-president or three managers. He believes that he has probability 0.6 of successfully recruiting the vicepresident candidate and probability 0.8 of successfully recruiting each of the manager candidates. The three
Everyone gets audited. Wallen Accounting Services specializes in tax preparation for individual tax returns. Data collected from past records reveals that 9% of the returns prepared by Wallen have been selected for audit by the Internal Revenue Service. Today,Wallen has six new customers. Assume
I’ll switch! In a 2007 buying-intention survey, Goldman Sachs found that71%of the respondents indicated interest in buying an Apple mobile phone. Of the respondents interested in buying an Apple mobile phone, 15% indicated that they would switch cellular carriers to get an Apple mobile phone.
Demographics in an SRS. The Census Bureau reports that 27% of California residents are foreign-born. Suppose that you choose three Californians at random, so that each has probability 0.27 of being foreign-born and the three are independent of each other. Let W be the number of foreign-born people
Using Internet sources. Internet sites often vanish or move, so that references to them can’t be followed. In fact, 13% of Internet sites referenced in major scientific journals are lost within two years after publication.(a) If a paper contains seven Internet references, what is the probability
Caffeine in the diet. Common sources of caffeine are coffee, tea, and cola drinks.Suppose that 55% of adults drink coffee 25% of adults drink tea 45% of adults drink cola and also that 15% drink both coffee and tea 5% drink all three beverages 25% drink both coffee and cola 5% drink only tea Draw a
Prosperity and education. Call a household prosperous if its income exceeds$100,000. Call the household educated if the householder completed college. Select an American household at random, and let A be the event that the selected household is prosperous and B the event that it is educated.
Bright lights? A string of holiday lights contains 20 lights. The lights are wired in series, so that if any light fails the whole string will go dark. Each light has probability 0.02 of failing during a 3-year period. The lights fail independently of each other. What is the probability that the
Failing to detect drug use. In Example 5.4, we considered how drug tests can indicate illegal drug use when no illegal drugs were actually used. Consider nowanother type of false test result. Suppose an employee is suspected of having used an illegal drug and is given two tests that operate
Misleading r´esum´es. For more than two decades, Jude Werra, president of an executive recruiting firm, has tracked executive r´esum´es to determine the rate of misrepresenting education credentials and/or employment information. On a biannual basis, Werra reports a now nationally recognized
College-educated part-time workers? For people aged 25 years or older, government data show that 34% of employed people have at least 4 years of college and that 20% of employed people work part-time. Can you conclude that because(0.34)(0.20) = 0.068 about 6.8% of employed people aged 25 years or
High school rank. Select a first-year college student at random and ask what his or her academic rank was in high school. Here are the probabilities, based on proportions from a large sample survey of first-year students:Rank: Top 20% Second 20% Third 20% Fourth 20% Lowest 20%Probability: 0.41 0.23
Infinite sample spaces, Part II (optional). Randomly choose one 30-milliliter (ml) ink cartridge from the lot of cartridges produced at your plant in the last hour. Let W be the actual amount of ink in the cartridge. The cartridges are supposed to contain 30 ml of ink but are designed to hold up to
Infinite sample spaces, Part I (optional). Buy one share of Apple Computer (AAPL) stock, and let Y be the number of days until the closing market value of your share is double what you initially paid for the share.(a) Give the sample space S for the possible values of Y .(b) Is Y a discrete random
Finite sample spaces. Choose one employee of your company at random and let X be the number of years that person has been employed by the company (rounded to the nearest year).(a) Give a reasonable sample space S for the possible values of X.(b) Is X a discrete random variable or a continuous
The risk of selling insurance, continued. The risk of insuring one person’s life is reduced if we insure many people.Use the result of the previous exercise and the rules for means and variances to answer the following questions.(a) Suppose that we insure two 21-year-old males, and that their
The risk of selling insurance. We have seen that the risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is (the larger σ is), the riskier the investment.We can measure the great risk of insuring one person’s life in
More about life insurance. It would be quite risky for you to insure the life of a 21-year-old friend under the terms of Exercise 4.151. There is a high probability that your friendwould live and you would gain $1250 in premiums. But if he were to die, you would lose almost $100,000. Explain
Life insurance. A life insurance company sells a term insurance policy to a 21-year-old male that pays $100,000 if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250 each
More about fire insurance. In fact, the insurance company sees that in the entire population of homeowners, the mean loss from fire is μ=$400 and the standard deviation of the loss is σ =$300. The distribution of losses is strongly right-skewed: many policies have $0 loss, but a few have large
Fire insurance. An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year is μ = $400 per person. (Most of us have no loss, but a few lose their homes. The $400 is the average loss.)The company plans to sell fire insurance for $400 plus
Simulating a mean. One consequence of the law of large numbers is that once we have a probability distribution for a random variable, we can find its mean by simulating many outcomes and averaging them. The law of large numbers says that if we take enough outcomes, their average value is sure to
Sadie’s portfolio. Continuing the previous exercise, for each of the three investment strategies, 90% of the time the mean monthly return x based on 12 months will be greater than what value?
Sadie’s portfolio. Examples 4.15 (page 249) and 4.20(page 256) provide the means and standard deviations of monthly returns for Dell, Apple, and Sadie’s portfolio (60% Dell and 40%Apple). As noted in Exercise 4.112 (page 275), stock returns follow a distribution that is slightly non-Normal. It
Risk pooling in a supply chain. Example 4.19 (page 254) shows that the standard deviation of demand on the centralized warehouse is less than the sum of standard deviations of the demands on each of the decentralized warehouses. The smaller the standard deviation of demand on the centralized
Risk pooling in a supply chain. Example 4.19(page 254) compares a decentralized versus a centralized inventory system as it ultimately relates to the amount of safety stock (extra inventory over and above mean demand) held in the system. Suppose that the CEO of ElectroWorks requires a 99%customer
A grade distribution. North Carolina State University has posted the grade distributions for its courses online. You can find that the distribution of grades in a large section of Accounting 210 in past semester to be:Grade: A B C D F Probability: 0.18 0.32 0.34 0.09 0.07(a) Verify that this is a
How many people in a car? A study of rush-hour traffic in San Francisco counts the number of people in each car entering a freeway at a suburban interchange. Suppose that this count has mean 1.5 and standard deviation 0.75 in the population of all cars that enter at this interchange during rush
Weights of eggs. The weight of the eggs produced by a certain breed of hen is Normally distributed with mean 65 grams(g) and standard deviation 5 g. Think of cartons of such eggs as SRSs of size 12 from the population of all eggs. What is the probability that the weight of a carton falls between
An IQ test. The Wechsler Adult Intelligence Scale(WAIS) is a common “IQ test” for adults. The distribution of WAIS scores for persons over 16 years of age is approximately Normal with mean 100 and standard deviation 15.(a) What is the probability that a randomly chosen individual has a WAIS
Nonstandard dice. You have two balanced, six-sided dice. One is a standard die, with faces having 1, 2, 3, 4, 5, and 6 spots. The other die has three faces with 0 spots and three faces with 6 spots. Find the probability distribution for the total number of spots Y on the up-faces when you roll
Rolling dice. Figure 4.2 (page 225) shows the possible outcomes for rolling two dice. If the dice are carefully made, each of these 36 outcomes has probability 1/36. The outcome of interest to a gambler is the sum of the spots on the two up-faces. Call this random variable X. Example 4.5 shows that
Classifying occupations. Choose an American worker at random and assign his or her occupation to one of the following classes. These classes are used in government employment data.A Managerial and professional B Technical, sales, administrative support C Service occupations D Precision production,
Age profile of social networkers. Millions of people regularly use social networking services on the Internet to communicate, sharing interests and activities. In a 2008 comprehensive study by the Web search firm RapLeaf, the age distribution of MySpace, Facebook, and LinkedIn users were as
Global warming in Canadian minds. Given Canada’s northern location and the shrinking of the ice caps (both Canadian and Arctic), global-warming issues are reported daily in the media. In a nationwide random sample of 1000 Canadians(September 29, 2008), the public opinion research firm Compas
Predicting the ACC champion. Las Vegas Zeke, when asked to predict the Atlantic Coast Conference basketball champion, follows the modern practice of giving probabilistic predictions.He says, “North Carolina’s probability of winning is twice Duke’s. North Carolina State and Virginia each have
Who gets promoted? Exactly one of Brown, Chavez, and Williams will be promoted to partner in the law firm that employs them all. Brown thinks that she has probability 0.25 of winning the promotion and that Williams has probability 0.2.What probability does Brown assign to the outcome that Chavez is
Customer backgrounds. A company that offers courses to prepare would-be MBA students for the GMAT examination has the following information about its customers: 20% are currently undergraduate students in business;15%are undergraduate students in other fields of study; 60% are college graduates who
Weights of bags of potato chips. In Example 4.8(page 233), the weight of a single 9-ounce bag of potato chips has an N(9.12, 0.15) distribution. Consider taking a sample of bags of chips and calculating the average weight of the sample.(a) What is the probability of a 3-bag average falling between
Weights of bags of potato chips. In Example 4.8(page 233), the probability of a single randomly chosen 9-ounce bag weighing between 9.33 and 9.45 ounces was calculated to be 0.0669. In the setting of the previous exercise, Samuel regularly samples 3 bags of chips and calculates their average
Weights of bags of potato chips. In Example 4.8(page 233), the weight of a single 9-ounce bag of potato chips has an N(9.12, 0.15) distribution. Samuel works on the production line that manufactures these bags of chips. He weighs 3 randomly chosen bags and records their mean.(a) What is the
Insuring an asset. Continue analyzing the insurance scenario described in the previous two exercises. Consider the value of holding the insurance policy and owning the house, X + Y . Let W = X + Y in this exercise.(a) Write down the probability distribution of W in the form of a table. What do you
Insuring an asset. Continue analyzing the insurance scenario described in the previous exercise.(a) Calculate the mean and standard deviation of X, the value of the insurance policy at the end of the next year.(b) Calculate the mean and standard deviation of Y , the value of the house at the end of
Insuring an asset. Consider an insurance policy that will pay the holder of the policy the market price of her house if the house is completely destroyed by fire. The market price of a particular house is set at $220,000. The probability of this house being completely destroyed by fire during the
Budgeting for expenses. Weekly postage expenses for your company have a mean of $312 and a standard deviation of$58. Your company has allowed $400 for postage per week in its budget.(a) What is the approximate probability that the average weekly postage expense for the past 52 weeks will exceed the
Auto accidents. The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. This distribution takes only whole-number values, so it is certainly not Normal.(a) Let x be the mean number of accidents per week at the intersection during a year (52
Manufacturing defects. Newly manufactured automobile radiators may have small leaks. Most have no leaks, but some have 1, 2, or more. The number of leaks in radiators made by one supplier has mean 0.15 and standard deviation 0.4. The distribution of number of leaks cannot be Normal because only
The business of insurance. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn
The cost of Internet access. The amount that households pay service providers for access to the Internet varies quite a bit, but the mean monthly fee is $28 and the standard deviation is$10. The distribution is not Normal: many households pay about$10 for limited dial-up access or about $25 for
Supplier delivery times. Supplier on-time delivery performance is critical to enabling the buyer’s organization to meet its customer service commitments. Therefore, monitoring supplier delivery times is critical. Based on a great deal of historical data, a manufacturer of personal computers finds
Safe flying weight. In response to the increasing weight of airline passengers, the Federal Aviation Administration told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary: the FAA gave a mean but not a standard
Making auto parts. An automatic grinding machine in an auto parts plant prepares axles with a target diameterμ = 40.125 millimeters (mm). The machine has some variability, so the standard deviation of the diameters is σ =0.002 mm. A sample of 4 axles is inspected each hour for process control
Dust in coal mines. A laboratory weighs filters from a coal mine to measure the amount of dust in the mine atmosphere.Repeated measurements of the weight of dust on the same filter vary Normally with standard deviation σ = 0.08 milligram(mg) because the weighing is not perfectly precise. The dust
Ages of Canadian farmers. Every five years, Statistics Canada conducts a Census of Agriculture. The target of the census is all farms selling one or more agricultural products. In 2006, Statistics Canada reported that the average age of a farmer on Prince Edward Island was 51.4 years old, while the
Business employees. There are more than 5 million businesses in the United States. The mean number of employees in these businesses is about 19. A university selects a random sample of 100 businesses in North Dakota and finds that they average about 14 employees. Is each of the bold numbers a
What Americans buy. In a story on consumer spending, Time magazine reports that Americans buy 34 Porsche 911s and 88,163 Apple iPods each day. Is each of the bold numbers a parameter or a statistic?
Legal music downloads. According to Apple, Inc. press reports, the iTunes Music Store sold 500,000,000 songs between July 17, 2005, and February 24, 2006. In an analysis of over 2 million credit and debit card transactions, Forrester Research(Forrester.com) calculated that “iTunes households”
Returns on stocks. The distribution of annual returns on common stocks is roughly symmetric, but extreme observations are somewhat more frequent than in Normal distributions. Because the distribution is not strongly non-Normal, the mean return over a number of years is close to Normal.
Flaws in carpets. The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per square yard. The population distribution cannot be Normal, because a count takes only whole-number values. An inspector samples 200
ACT scores. The scores of students on the ACT college entrance examination in a recent year had the Normal distribution with meanμ = 18.6 and standard deviationσ = 5.9.(a) What is the probability that a single student randomly chosen from all those taking the test scores 21 or higher?(b) Now take
Measuring blood cholesterol. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of youth of ages 13 to 16 years. The researchers will report the mean x from their sample as an estimate of the mean cholesterol level μ in this population.(a) Explain to someone
Measurements on the production line. Sodium content (in milligrams) is measured for bags of potato chips sampled from a production line. The standard deviation of the sodium content measurements is σ = 10 mg. The sodium content is measured 3 times and the mean x of the 3 measurements is
Generating a sampling distribution. Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the sizes of 10 medium-sized businesses where size is measured in terms of the number of employees. For convenience, the 10
The law of large numbers. Figure 4.2 (page 225) shows the 36 possible outcomes of rolling two dice and counting the spots on the up-faces. These 36 outcomes are equally likely. You can calculate that the mean for the sum of the two up-faces isμ = 7. This is the population mean μ for the idealized
The “law of averages.” The baseball Hall of Famer Tony Gwynn got a hit about 34% of the time over his 20-year career. After he failed to hit safely in six straight at-bats, the TV commentator said, “Tony is due for a hit by the law of averages.” Is that right? Why?
Help this man.(a) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds and bets heavily on red at the next spin. Asked why, he says that “red is hot” and that the run of reds is likely to continue. Explain to the
Playing the numbers. The numbers racket is a well-entrenched illegal gambling operation in most large cities. One version works as follows. You choose one of the 1000 three-digit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one three-digit number is
Successive averages. Figure 4.15 shows how the mean of n observations behaves as we keep adding more observations to those already in hand. The first 10 observations are given in Example 4.21. Demonstrate that you grasp the idea of Figure 4.15: find the mean of the first one, then two, three, four,
Comparing computers. Pfeiffer Consulting, a technology consulting group, designed benchmark tests to compare the speed with which computer models complete a variety of tasks. Pfeiffer announced that the mean completion time system startup was 56.6 seconds for 2.0-GHz Power Mac G5 models and 30.1
Irrational decision making? The psychologist Amos Tversky did many studies of our perception of chance behavior.In its obituary of Tversky (June 6, 1996), the New York Times cited the following example.(a) Tversky asked subjects to choose between two public health programs that affect 600 people.
Making glassware. In a process for manufacturing glassware, glass stems are sealed by heating them in a flame. The temperature of the flame varies a bit. Here is the distribution of the temperature X measured in degrees Celsius:Temperature: 540◦ 545◦ 550◦ 555◦ 560◦Probability: 0.1 0.25
A hot stock. You purchase a hot stock for $1000. The stock either gains 30% or loses 25% each day, each with probability 0.5. Its returns on consecutive days are independent of each other. This implies that all four possible combinations of gains and losses in two days are equally likely, each
Perfectly correlated variables (optional). We know that variances add if the random variables involved are uncorrelated(ρ = 0), but not otherwise. The opposite extreme is perfect positive correlation (ρ = 1). Show by using the general addition rule for variances that in this case the standard
Larger portfolios (optional). Portfolios often contain more than two investments. The rules for means and variances continue to apply, though the arithmetic gets messier.A portfolio containing proportions a of Magellan Fund, b of Energy Fund, and c of Japan Fund has return R = aW +bX +cY .Becausea,
More on diversification. Diversification works better when the investments in a portfolio have small correlations.To demonstrate this, suppose that returns on Magellan Fund and Japan Fund had the means and standard deviations we have given but were uncorrelated (ρWY = 0). Show that the standard
Diversification. Many advisers recommend using roughly 20% foreign stocks to diversify portfolios of U.S.stocks. Michael owns Fidelity Magellan Fund, which concentrates on stocks of large American companies. He decides to move to a portfolio of 80% Magellan and 20% Fidelity Japan Fund. Show that
Vegas versus actual NFL point spreads. Based on all 224 NFL games in 2007, the meanVegas point spread (=Underdog−Favorite) is −5.82 and the standard deviation is 3.94. The actual point spreads have mean −6.69 and standard deviation 12.25.The correlation between Vegas and actual point spreads
SAT combined score. Exercise 4.27 (page 234) reported that scores on the SAT Math Reasoning test are approximately Normal with mean 515 points and standard deviation 116 points.The scores on the Verbal Reasoning test are also approximately Normal but with mean 502 and standard deviation 112. The
Study habits. The academic motivation and study habits of female students as a group are better than those of males. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures these factors. The distribution of SSHA scores among the women at a college has mean 120 and
Gain Communications, continued. Redo Exercise 4.89 assuming correlation 0.01 between civilian and military sales.Redo the exercise assuming correlation 0.99. Comment on the effect of small and large correlations on the uncertainty of Gain’s sales projections.
Gain Communications. Examples 4.16 and 4.17 concern a probabilistic projection of sales and profits by an electronics firm, Gain Communications. The mean and variance of military sales X appear in Example 4.17 (page 252). You found the mean and variance of civilian sales Y in Exercise 4.61(page
Combining measurements. You have two scales for measuring weight. Both scales give answers that vary a bit in repeated weighings of the same item. If the true weight of an item is 2 grams(g), the first scale produces readings X that have mean 2.000 g and standard deviation 0.002 g. The second
Get the mean you want. Here is a simple way to create a random variable X that has mean μ and standard deviation σ: X takes only the two values μ − σ and μ + σ, each with probability 0.5. Use the definition of the mean and variance for discrete random variables to show that X does have mean
are independent. Find the standard deviation of the time required to complete the process.
A chemical production process. The times for the two reactions in the chemical production process described in Exercise
Time-and-motion studies. Find the standard deviation of the time required for the two-step assembly operation studied in Exercise 4.83, assuming that the study shows the two times to be independent. Redo the calculation assuming that the two times are dependent, with correlation 0.3. Can you
A chemical production process. Laboratory data show that the time required to complete two chemical reactions in a production process varies. The first reaction has a mean time of 40 minutes and a standard deviation of 2 minutes; the second has a mean time of 25 minutes and a standard deviation of
Time-and-motion studies. A time-and-motion study measures the time required for an assembly-line worker to perform a repetitive task. The data show that the time required to bring a part from a bin to its position on an automobile chassis varies from car to car with mean 11 seconds and standard
Independent random variables? In which of the following games of chance would you be willing to assume independence of X and Y in making a probability model? Explain your answer in each case.(a) In blackjack, you are dealt two cards and examine the total points X on the cards (face cards count 10
Independent random variables? For each of the following situations, would you expect the random variables X and Y to be independent? Explain your answers.(a) X is the rainfall (in inches) on November 6 of this year and Y is the rainfall at the same location on November 6 of next year.(b) X is the
Pick 3 once more. The Tri-State Pick 3 lottery game offers a choice of several bets. You choose a three-digit number. The lottery commission announces the winning three-digit number, chosen at random, at the end of each day. The “box” pays $83.33 if the number you choose has the same digits as
Gain Communications. Refer to the previous exercise and perform the same Excel or Minitab experiment on the civilian sales distribution shown in Example 4.16.
Gain Communications. Example 4.16 provides the probability distribution for military sales X and the mean of X. From this section, we learned that the mean of a probability distribution describes the long-run average outcome. In this exercise, you will explore this concept using technology.•
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