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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
Leaking gas tanks. Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks leak.You examine 15 tanks chosen at random, independently of each other.(a) What is the mean number of leaking tanks in such samples of 15?(b) What is
Stealing software. Employees sometimes install on their home computers software that was purchased by their employer for use on their work computers. For most commercial software packages, this is illegal. Suppose that 5% of all employees at a large corporation have illegally installed corporate
Who buys iMacs? When the iMac computer was first introduced by Apple Computer, it quickly became one of the company’s best-selling products. The iMac was particularly aimed at first-time computer buyers. Approximately 5 months after the introduction of the iMac, Apple reported that 32% of iMac
Screening job applicants. A company retains a psychologist to assess whether job applicants are suited for assembly-line work. The psychologist classifies applicants as A (well suited), B (marginal), or C (not suited). The company is concerned about event D: an employee leaves the company within a
More on race and ethnicity. Use the information in Exercise 5.113 to answer these questions.(a) What is the probability that a randomly chosen American is Hispanic?(b) You know that the person chosen is black. What is the conditional probability that this person is Hispanic?
More on teenage drivers. Use your work from Exercise 5.112 to find the percent of A students who are involved in accidents. (Start by expressing this as a conditional probability.)
Race and ethnicity. Annually, the U.S. Census Bureau provides estimates of a variety of demographic characteristics of the U.S. population. In terms of categorizing people by race, the bureau divides the population into those of “Hispanic” origin and those not.Within these two origin
Teenage drivers. An insurance company has the following information about drivers aged 16 to 18 years: 20% are involved in accidents each year; 10% in this age group are A students;among those involved in an accident, 5% are A students.(a) Let A be the event that a young driver is an A student and
The geometric distributions. In some manufacturing environments, all manufactured items are inspected, and when the first defect is encountered, the operation is halted for investigation of possible causes. Suppose that the rate of defective items is 1% and that the quality of each item is
Testing for HIV, continued. The previous exercise gives data on the results of EIA tests for the presence of antibodies to HIV. Repeat part (c) of that exercise for two different populations:(a) Blood donors are prescreened for HIV risk factors, so perhaps only 0.1% (0.001) of this population
Testing for HIV. Enzyme immunoassay (EIA) tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. Here are approximate probabilities of positive
Canadian children and family structure. It is common for governments to gather data on family structure in the population.Information on family structure distribution is important because of the known associations between family structure and child well-being, behavior, and general success in life.
Applying for student financial aid. To determine eligibility for federal student financial aid, students (undergraduate and graduate) must fill out a form knownas “The Free Application for Federal Student Aid” (FAFSA). Many students are unaware of the FAFSA, while other students learn about it
Income and savings. A sample survey chooses a sample of households and measures their annual income and their savings. Some events of interest are A = the household chosen has income at least $100,000 C = the household chosen has at least $50,000 in savings Based on this sample survey, we estimate
Medical risks. You have torn a tendon and are facing surgery to repair it. The surgeon explains the risks to you: infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occur together in 1%. What percent of these operations succeed and are free from
Playing the slots. Slot machines are now video games, with winning determined by electronic random number generators.In the old days, slot machines worked like this: you pull the lever to spin three wheels; each wheel has 20 symbols, all equally likely to show when the wheel stops spinning; the
Human spell-checker. Spelling errors in a text can be either nonword errors or word errors. Nonword errors make up 25% of all errors. A human proofreader will catch 90% of nonword errors and 70% of word errors. What percent of all errors will the proofreader catch? (Draw a tree diagram to organize
Motor vehicle sales. Using the information in Exercise 5.100, answer these questions.(a) Given that a vehicle is imported, what is the conditional probability that it is a car?(b) Are the events “vehicle is a car” and “vehicle is imported”independent? Justify your answer.
Using a mailing list. The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 70%males and 30% females. The agency calls 30 people chosen at random from its list.(a) What is the probability that 20 of the 30 are men? (Use the binomial probability formula.)(b) What
Motor vehicle sales. Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and as either domestic or imported. From January to October 2008, 48% of vehicles sold were light trucks, 75% were domestic, and 40% were domestic light trucks.(a) Draw a Venn
Calls to 911 system. The City of Chicago reports that on an average day 15,000 emergency calls come into the city’s 911 system.15 Assuming a Poisson distribution with this reported mean rate, use the Normal distribution to find the range in which we would expect 99.7% of the calls to fall.
Initial public offerings. The number of companies making their initial public offering of stock (IPO) can be modeled by a Poisson distribution with a mean of 15 per month.(a) What is the probability of fewer than 3 IPOs in a month?(b) What is the probability of fewer than 15 IPOs in a month?(c)
Credit card manufacturing. Large sheets of plastic are cut into smaller pieces to be pressed into credit cards. One manufacturer uses sheets of plastic known to have approximately 2.3 defects per square yard. The number of defects can be modeled as a Poisson random variable X.(a) What is the
Web site hits. A “hit” for a Web site is a request for a file from the Web site’s server computer. Some popular Web sites have thousands of hits per minute. One popular Web site boasts an average of 6500 hits per minute between the hours of 9 A.M.and 6 P.M. Some weaker software packages will
Calling tech support. The number of calls received between 8 A.M. and 9 A.M. by a software developer’s technical support line has a Poisson distribution with a mean of 14.(a) What is the probability of at least 5 calls between 8 A.M. and 9 A.M.?(b) What is the probability of at least 5 calls
Flaws in carpets. Flaws in carpet material follow the Poisson model with mean 1.6 flaws per square yard. An inspector examines 100 randomly selected square yard specimens of the material, records the number of flaws found in each specimen, and calculates x, the average number of flaws per square
Work-related deaths. Based on 2007 government data, work-related deaths in the United States have a mean of 15 per day. Suppose the count of work-related deaths per day follows an approximate Poisson distribution.(a) What is the standard deviation for daily work-related deaths?(b) What is the
Too much email? According to email logs, one employee at your company receives an average of 110 emails per week.Suppose the count of emails received can be adequately modeled as a Poisson random variable.(a) What is the distribution of the number of emails in a twoweek period?(b) What is the
Traffic model. The number of vehicles passing a particular mile marker during 15-minute units of time can be modeled as a Poisson random variable. Counting devices show that the average number of vehicles passing the mile marker every 15 minutes is 48.7.(a) What is the probability of 50 or more
Too much email? According to email logs, one employee at your company receives an average of 110 emails per week.Suppose the count of emails received can be adequately modeled as a Poisson random variable.(a) What is the probability of this employee receiving exactly 110 emails in a given week?(b)
Switchboard errors. At a large software company, calls coming to the general contact phone number are answered by a switchboard operator. The operator’s job is to manually forward calls to the desired party within the company. The switchboard operator’s past performance indicates that he
Bad checks. Duck Worth Wearing, a children’s consignment shop, accepts cash, check, and credit card for payment.About 30% of all transactions are paid by check. Duck Worth Wearing anticipates only 3 bad checks per month, on average.Assume that the number of bad checks per month can be modeled as
Hospitalization claims. A national home improvement and hardware chain provides health insurance to its full-time employees.On the basis of past experience with companies similar to this one, the insurance carrier for this chain of stores expects 6 hospitalizations per month, on average. Employee
Howmany calls? Calls to the customer service department of a cable TV provider are made randomly and independently at a rate of 11 per minute. The company has a staff of 20 customer service specialists who handle all the calls. Assume that none of the specialists are on a call at this moment and
A safety initiative. This year, a “safety culture change” initiative attempts to reduce the number of accidents at the plant described in the previous exercise. There are 66 reportable accidents during the year. Suppose that the Poisson distribution of the previous exercise continues to
Industrial accidents. A large manufacturing plant has averaged 7 “reportable accidents” per month. Suppose that accident counts over time follow a Poisson distribution with mean 7 per month.(a) What is the probability of exactly 7 accidents in a month?(b) What is the probability of 7 or fewer
Ichiro Suzuki’s hits. Using the continuity correction discussed in Exercise 5.82, redo part (b) of Exercise 5.78. How does this value compare with the original approximation and the exact answer of part (d)?
Continuity correction. We have found that when the sample size n is large enough, the Normal distribution well approximates the binomial distribution. There is, however, a fine-tuning that can be done to make the approximation even better. The fine-tuning, known as “continuity correction,”
Checking for survey errors. One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known facts about the population. About 12% of American adults are black.The number X of blacks in a random sample of 1500 adults
Are we shipping on time? Your mail-order company advertises that it ships 90% of its orders within three working days.You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time.(a) If the company really ships
Planning a survey. You are planning a sample survey of small businesses in your area. You will choose an SRS of businesses listed in the telephone book’s Yellow Pages. Experience shows that only about half the businesses you contact will respond.(a) If you contact 150 businesses, it is reasonable
Ichiro Suzuki’s hits. In 2004, Ichiro Suzuki of the Seattle Mariners had 262 hits, a new Major League record breaking an 84-year-old mark held by George Sisler of the St. Louis Browns, who had 257 hits. “I just hope people realize the monumental effort it took to surpass this record, which has
Multiple-choice tests. Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) Answers to different
Lie detectors. A federal report finds that lie detector tests given to truthful persons have probability about 0.2 of suggesting that the person is deceptive.13(a) A company asks 12 job applicants about thefts from previous employers, using a lie detector to assess their truthfulness.Suppose that
Random stock prices. A believer in the “random walk”theory of stock markets thinks that an index of stock prices has probability 0.65 of increasing in any year. Moreover, the change in the index in any given year is not influenced by whether it rose or fell in earlier years. Let X be the number
Generic brand soda. In a taste test of a generic soda versus a brand name soda, 25% of tasters can distinguish between the colas. Twenty tasters are asked to take the taste test and guess which cup contains the brand name soda. The tests are done independently in separate locations, so that the
Unmarried women. Among employed women, 25% have never been married. Select 10 employed women at random.(a) The number in your sample who have never been married has a binomial distribution. What are n and p?(b) What is the probability that exactly 2 of the 10 women in your sample have never been
Random digits. Each entry in a table of random digits like Table B has probability 0.1 of being a 0, and digits are independent of each other.(a) What is the probability that a group of five digits from the table will contain at least one 0?(b) What is the mean number of 0s in lines 40 digits long?
Binomial setting? In each of the following cases, decide whether or not a binomial distribution is an appropriate model, and give your reasons.(a) A firm uses a computer-based training module to prepare 20 machinists to use new numerically controlled lathes. The module contains a test at the end of
Binomial setting? In each situation below, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case.(a) An auto manufacturer chooses one car from each hour’s production for a detailed quality inspection. One variable recorded is the
Internet postings. Suppose (as is roughly true) that 20%of all Internet users have posted photos online. A sample survey interviews an SRS of 1555 Internet users.(a) What is the actual distribution of the number X in the sample who have posted photos online?(b) What is the probability that 300 or
Carpooling stats. Although cities encourage carpooling to reduce traffic congestion, most vehicles carry only one person.For example, 70% of vehicles on the roads in the Minneapolis–St. Paul metropolitan area are occupied by just the driver.(a) If you choose 10 vehicles at random, what is the
Web site traffic. Suppose that 50% of male Internet users aged 18 to 34 have visited an auction site at least once in the past month.(a) If you interview 12 at random, what is the mean of the count X who have visited an auction site? What is the mean of the proportion ˆp in your sample who have
Web site traffic. What kinds ofWeb sites do males aged 18 to 34 visit most often? Pornographic sites take first place,12 but about 50% of male Internet users in this age group visit an auction site such as eBay at least once a month. Interview a random sample of 12 male Internet users aged 18 to
The effect of sample size. TheSRSof size 200 described in the previous exercise finds that 100 of the 200 respondents are concerned about nutrition.We wonder if this is reason to conclude that the percent in your area is higher than the national 40%.(a) Find the probability that X is 100 or larger
Restaurant survey. Return to the survey described in Exercise 5.60. You plan to use random digit dialing to contact an SRS of 200 households by telephone rather than just 20.(a) What are the mean and standard deviation of the number of nutrition-conscious people in your sample if p = 0.4 is
if the university’s claim is true.(b) Find the standard deviation σ of the count X.(c) Suppose that the 20 players came from a population of which p = 0.9 graduated.What is the standard deviation σ of the count of graduates? If p = 0.99, what isσ? What does your work show about the behavior of
Do our athletes graduate?(a) Find the mean number of graduates out of 20 players in the setting of Exercise
Hispanic representation.(a) What is the mean number of Hispanics on randomly chosen committees of 15 workers in Exercise 5.58?(b) What is the standard deviation σ of the count X of Hispanic members?(c) Suppose that 10% of the factory workers were Hispanic. Then p = 0.1. What isσ in this case?
Restaurant survey. You operate a restaurant. You read that a sample survey by the National Restaurant Association shows that 40% of adults are committed to eating nutritious food when eating away from home. To help plan your menu, you decide to conduct a sample survey in your own area. You will use
Do our athletes graduate? A university claims that 80% of its basketball players get degrees. An investigation examines the fate of all 20 players who entered the program over a period of several years that ended six years ago. Of these players, 11 graduated and the remaining 9 are no longer in
Hispanic representation. A factory employs several thousand workers, of whom 30% are Hispanic. If the 15 members of the union executive committee were chosen from the workers at random, the number of Hispanics on the committee would have the binomial distribution with n = 15 and p = 0.3.(a) What is
Misleading r´esum´es. In Exercise 5.3 (page 290), we found that 15.9% of executive job applicants lied on their r´esum´es. Suppose an executive job hunter randomly selects 5 r´esum´es from an executive job applicant pool. Let X be the number of misleading r´esum´es found in the sample. So X
Teaching office software. A company uses a computer-based system to teach clerical employees new office software. After a lesson, the computer presents 10 exercises.The student solves each exercise and enters the answer. The computer gives additional instruction between exercises if the answer is
Customer satisfaction calls. The service department of an automobile dealership follows up each service encounter with a customer satisfaction survey by means of a phone call. On a given day, let X be the number of customers a service representative has to call until a customer is willing to
Deliveries. A courier service audits the records of 100 next–business-morning(before 10 A.M.) deliveries; X is the number of on-time deliveries among them.
Inspecting final products. Final products are sometimes selected to go through a complete inspection before leaving the production facility. Suppose that 8% of all products made at a particular facility fail to conform to specifications. Furthermore, 55%of all nonconforming items are selected for
Successful bids, continued. Draw a Venn diagram that illustrates the relation between events A and B in Exercise 5.50.Write each of the following events in terms of A, B, Ac, and Bc.Indicate the events on your diagram and use the information in Exercise 5.50 to calculate the probability of each.(a)
Independence? In the setting of the previous exercise, are events A and B independent? Do a calculation that proves your answer.
Successful bids. 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 (event {A
Credit card defaults. The credit manager for a local department store discovers that 88% of all the store’s credit card holders who defaulted on their payments were late (by a week or more) with two or more of their monthly payments before failing to pay entirely (defaulting). This prompts the
Sales to women. In the setting of Exercise 5.46, what percent of sales are made to women? (Write this as a conditional probability.)
Success on theGMAT. In the setting of Exercise 5.45, what percent of the customers who score at least 600 on the GMAT are undergraduates? (Write this as a conditional probability.)
Telemarketing. A telemarketing company calls telephone numbers chosen at random. It finds that 70% of calls are not completed (the party does not answer or refuses to talk), that 20% result in talking to a woman, and that 10% result in talking to a man. After that point, 30% of the women and 20% of
Preparing for the GMAT. A company that offers courses to prepare would-be MBA students for the GMAT examination finds that 40% of its customers are currently undergraduate students and60%are college graduates. After completing the course, 50% of the undergraduates and 70% of the graduates achieve
Employment status, continued.(a) What is the probability that a randomly chosen person 25 years of age or older is in the labor force?(b) If you know that the person chosen is a college graduate, what is the conditional probability that he or she is in the labor force?(c) Are the events “in the
Employment status. As noted in Example 5.6 (page 295), in the language of government statistics, you are “in the labor force” if you are available for work and either working or actively seeking work. The unemployment rate is the proportion of the labor force (not of the entire population) who
Classifying occupations. Exercise 4.137 (page 280) gives the probability distribution of the gender and occupation of a randomly chosen American worker. Use this distribution to answer the following questions.(a) Given that the worker chosen holds a managerial (Class A)job, what is the conditional
A little geometry. Choose a point at random in the square with sides 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. This means that the probability that the point falls in any region within the square is equal to the area of that region. Let X be the x coordinate and Y the y coordinate of the point chosen. Find
College degrees. Exercise 5.28 (page 299) gives the counts(in thousands) of earned degrees in the United States in a recent year. Use these data to answer the following questions.(a) What is the probability that a randomly chosen degree recipient is a man?(b) What is the conditional probability
High school baseball players. It is estimated that 56% of MLB players have careers of 3 or more years.9 Using the information in Example 5.10, determine the proportion of high school players expected to play 3 or more years in MLB.
High school football players. Using the information in Example 5.9, determine the proportion of high school football players expected to play professionally in the NFL.
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) What is the conditional probability that a student likes gospel music if we know that he or she likes
Income tax returns. In 2006, the Internal Revenue Service received 138,394,754 individual tax returns. Of these, 16,153,307 reported an adjusted gross income of at least $100,000, and 354,093 reported at least $1 million.(a) What is the probability that a randomly chosen individual tax return
Loan officer decision. Considering the information provided in the previous exercise, calculate P(O|D). Show your work. Also, express this probability in words in the context of the loan officer’s decision. If newinformation about the customer becomes available before the loan officer makes her
Loan officer decision. A loan officer is considering a loan request from a customer of the bank. Based on data collected from the bank’s records over many years, there is an 8% chance that a customer who has overdrawn an account will default on the loan. However, there is only a 0.6% chance that
Self-service gas pumps. In the setting of the previous exercise, what percent of customers who pay at least $30 pump premium gas? (Write this as a conditional probability and use your result from the previous exercise.)
Self-service gas pumps. At a self-service gas station, 40%of the customers pump regular gas, 35% pump midgrade, and 25% pump premium gas. Of those who pump regular, 30% pay at least $30. Of those who pump midgrade, 50% pay at least $30.And of those who pump premium, 60% pay at least $30. What is
PDA screens. A manufacturer of Personal Digital Assistants (PDAs) purchases screens from two different suppliers. The company receives 55% of its screens from Screensource and the remaining screens from Brightscreens. The quality of the screens varies between the suppliers: Screensource supplies 1%
Where to manufacture? Zipdrive, Inc., has developed a new high-capacity external drive to be used primarily by notebook and laptop users. The demand for the new product is uncertain but can be described as “high” or “low” in any one year. After 4 years, the product is expected to be
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 householder at random, and let A be the event that the selected household is prosperous and B the event that the householder is
College degrees. Here are the counts (in thousands) of earned degrees in the United States in the 2005–2006 academic year, classified by level and by the gender of the degree recipient:7 Bachelor’s Master’s Professional Doctorate Total Female 855 356 44 27 1282 Male 631 238 44 29 942 Total
Employment status. Use the two-way table in Example 5.6 to find these conditional probabilities.(a) P(employed | some college)(b) P(employed | bachelor’s degree or higher)(c) P(high school degree | employed)(d) P(unemployed | bachelor’s degree or higher)(e) P(bachelor’s degree or higher |
Buying from Japan. Functional Robotics Corporation buys electrical controllers from a Japanese supplier. The company’s treasurer thinks that there is probability 0.4 that the dollar will fall in value against the Japanese yen in the next month.The treasurer also believes that if the dollar falls
Woman managers. Choose an employed person at random. Let A be the event that the person chosen is a woman, and B the event that the person holds a managerial or professional job. Government data tell us that P(A) = 0.46 and the probability of managerial and professional jobs among women is P(B | A)
Reward members. Refer to Exercise 5.20 to find that 290 passengers in the sample are reward members. Suppose two members are randomly selected from the 290 members. Find the probability that both members are very satisfied with the airline.
Playing the odds? A writer on casino games says that the odds against throwing an 11 in the dice game craps are 17 to 1. He then says that the odds against three 11s in a row are 17×17×17 to 1, or 4913 to 1.3(a) What is the probability that the sum of the up-faces is 11 when you throw two
Age effects in medical care. The type of medical care a patient receives may vary with the age of the patient. A large study of women who had a breast lump investigated whether or not each woman received a mammogram and a biopsy when the lump was discovered. Here are some probabilities estimated by
Is that independence? While preparing for the business statistics midterm, your study partner comments that if P(A) > 0, P(B) > 0, and P(A or B) = P(A)+ P(B), then the events A and B are independent. Is your study partner on mark or confused? Explain.
Customer satisfaction. An airline company conducts a customer satisfaction survey of its passengers to investigate differences between its mileage reward members and nonmembers.Here are the results for 500 randomly selected passengers:Reward Member Very satisfied Yes No Yes 203 129 No 87 81(a) What
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