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Statistics The Art And Science Of Learning From Data 4th Global Edition Alan Agresti, Christine A. Franklin, Bernhard Klingenberg - Solutions
6.36 Compatible donors Refer to the previous exercise. Let the random variable X = number of compatible donors.Check the conditions for X to be binomial by answering parts a–c.a. What constitutes a trial, how many trials are there, and how many outcomes does each trial have?b. Does each trial
6.35 Kidney transplants In kidney transplantations, compatibility between donor and receiver depends on such factors as blood type and antigens. Suppose that for a randomly selected donor from a large national kidney registry, there is a 10% chance that he or she is compatible with a specific
6.51 Binomial assumptions For the following random variables, check whether the conditions needed to use the binomial distribution are satisfied or not. Explaina. X = number of people suffering from a contagious disease in a family of 4, when the probability of catching this disease is 2% in the
6.52 Grandparents Let X = the number of living grandparents that a randomly selected adult American has. According to recent General Social Surveys, its probability distribution is approximately P102 = 0.71, P112 = 0.15, P122 = 0.09, P132 = 0.03, P142 = 0.02.a. Does this refer to a discrete or a
6.69 Global warming Suppose that weekly use of gasoline for motor vehicle travel by adults in North America has approximately a normal distribution with a mean of 20 gallons and a standard deviation of 6 gallons. Many people who worry about global warming believe that Americans should pay more
6.68 Used car deals Refer to the previous exercise. Suppose carmax.com decides to highlight any used car that is priced 1.5 standard deviations below the mean price as a special deal on its Web site.a. If the distribution of used-car prices is normal, what percentage of used Audi A4s will be
6.67 Used car prices Data from the Web site carmax.com compiled in July 2014 show that prices for used Audi A4 cars advertised on the Web site have a mean of $23,800 and a standard deviation of $4,380. Assume a normal distribution for the price.a. What percent of used Audi A4s cost more
6.66 Gestation times For 5459 pregnant women using Aarhus University Hospital in Denmark in a two-year period who reported information on length of gestation until birth, the mean was 281.9 days, with standard deviation 11.4 days. A baby is classified as premature if the gestation time is 258 days
6.65 Cloning butterflies The wingspans of recently cloned monarch butterflies follow a normal distribution with mean 9 cm and standard deviation 0.75 cm. What proportion of the butterflies has a wingspana. less than 8 cm?b. wider than 10 cm?c. between 8 and 10 cm?d. Ten percent of the butterflies
6.64 Female heights Female heights in North America follow a normal distribution with m = 65 inches and s = 3.5 inches. Find the proportion of females who area. under five feet.b. over six feet.c. between 60 and 70 inches.d. Repeat parts a–c for North American males, the heights of whom are
6.62 Boys and girls birth weight Exercise 6.28 mentioned that in the United States, birth weight of boys is approximately normal, with mean 3.41 kg and standard deviation 0.55 kg. For girls, the birth weight is also approximately normal with mean 3.29 kg and standard deviation 0.52 kg.a. A weight
6.61 Quartiles If z is the positive number such that the interval within z standard deviations of the mean contains 50% of a normal distribution, thena. Explain why this value of z is about 0.67.b. Explain why for any normal distribution the first and third quartiles equal m - 0.67s and m +
6.60 z-score and tail probabilitya. Find the z-score for the number that is less than only 1% of the values of a normal distribution. Sketch a graph to show where this value is.b. Find the z-scores corresponding to the (i) 90th and(ii) 99th percentiles of a normal distribution.
6.59 z-scores Find the z-score such that the interval within z standard deviations of the mean contains probability(a) 0.95 and (b) 0.99 for a normal distribution. Sketch the two cases on a single graph.
6.58 Normal probabilities For a normal distribution, find the probability that an observation isa. Within 1.96 standard deviations of the mean.b. More than 2.33 standard deviations from the mean.
6.57 Flyers’ insurance An insurance company sells a policy to airline passengers for $1. If a flyer dies on a given flight (from a plane crash), the policy gives $100,000 to the chosen beneficiary. Otherwise, there is no return.Records show that a passenger has about a one in a million chance of
6.56 Are you risk averse? You need to choose between two alternative programs for dealing with the outbreak of a deadly disease. In program 1, 200 people are saved. In program 2, there is a 2>3 chance that no one is saved and a 1>3 chance that 600 people are saved.a. Find the expected number of
6.55 Paths to school You have two possible paths to reach your school. Path A is 1.5 km long and path B is 1.3 km long. Each day, the probability you choose path A to reach school is 3/4 and this probability is 1/2 when you choose your path to return.a. List all the possible combinations of paths
6.54 Auctioning paintings A collector is interested in two paintings by the same artist available at an auction. She plans to bid $3,000 for the first painting auctioned off and$2,000 for the second. She estimates that the probability she will win the bid for the first painting (and spend$3,000) is
6.34 Relative grades The mean and standard deviation of the grades of a statistics course and an English course are 1m = 80, s = 4.52 and 1m = 85, s = 4.02, respectively.A student attends both the courses and scores 85 in statistics and 95 in English. Which grade is relatively better?Explain why.
5.81 Conference dinner Of the participants at a conference, 50% attended breakfast, 90% attended dinner, and 40%attended both breakfast and dinner. Given that a participant attended breakfast, find the probability that she also attended dinner.
5.61 DNA evidence compelling? DNA evidence can be extracted from biological traces such as blood, hair, and saliva. “DNA fingerprinting” is increasingly used in the courtroom as well as in paternity testing. Given that a person is innocent, suppose that the probability of his or her DNA
5.59 Was OJ actually guilty? Former pro football star O. J. Simpson was accused of murdering his wife. In the trial, a defense attorney pointed out that although Simpson had been guilty of earlier spousal abuse, annually only about 40 women are murdered per 100,000 incidents of partner abuse. This
5.58 More screening for breast cancer Refer to the previous exercise. For young women, the prevalence of breast cancer is lower. Suppose the sensitivity is 0.86 and the specificity is 0.88, but the prevalence is only 0.001.a. Given that a test comes out positive, find the probability that the woman
5.57 Mammogram diagnostics Breast cancer is the most common form of cancer in women, affecting about 10%of women at some time in their lives. There is about a 1%chance of having breast cancer at a given time (that is, P1S2 = 0.01 for the state of having breast cancer at a given time). The chance of
5.56 Rosencrantz and Guildenstern In the opening scene of Tom Stoppard’s play Rosencrantz and Guildenstern Are Dead, about two Elizabethan contemporaries of Hamlet, Guildenstern flips a coin 91 times and gets a head each time. Suppose the coin was balanced.a. Specify the sample space for 91 coin
5.55 A true coincidence of emergency E-Comm, British Columbia’s emergency communications center, provides communication services and support systems to two million residents of southwest British Columbia, Canada. On any given day, the probability a randomly selected resident decides to call
5.54 Monkeys typing Shakespeare Since events of low probability eventually happen if you observe enough trials, a monkey randomly pecking on a typewriter could eventually write a Shakespeare play just by chance. Let’s see how hard it would be to type the title of Macbeth properly.Assume 50 keys
5.53 Coincidence in your life State an event that has happened to you or to someone you know that seems highly coincidental (such as seeing a friend while on vacation).Explain why that event may not be especially surprising, once you think of all the similar types of events that could have happened
5.52 Horrible 11 on 9/11 The digits in 9/11 add up to 11 19 + 1 + 12, American Airlines flight 11 was the first to hit the World Trade Towers (which took the form of the number 11), there were 92 people on board 19 + 2 = 112, September 11 is the 254th day of the year 12 + 5 + 4 = 112, and there are
5.51 Failure and repair of photocopiers In a photocopy center, there are two small photocopiers, two medium photocopiers and one big photocopier. The probability that a small one fails and requires repairs is 0.1, a medium one fails and requires repairs is 0.08, and the probability that the big
5.50 Holes in one at Masters The Augusta National Golf Course in Augusta, Georgia, hosts the Masters Tournament each April. The course consists of four par 3s, ten par 4s, and four par 5s. The par 4s and par 5s are long enough so that no golfer has a realistic chance of getting a hole in one, but
5.49 Lots of pairs Show that with 25 students, there are 300 pairs of students who can have the same birthday. So it’s really not so surprising if at least two students have the same birthday. (Hint: You can pair 24 other students with each student, but how can you make sure you don’t count
5.48 Matching your birthday You consider your birth date to be special since it falls on January 1. Suppose your class has 25 students.a. Is the probability of finding at least one student with a birthday that matches yours greater, the same, or less than the probability found in Example 13 of a
5.47 Heart disease A particular heart disease is said to have a prevalence of 1/1000 in a specific population. In a sample of 50 people chosen randomly, what is the probability that at least two people have this disease?
5.46 Checking independence In each of three independent visits to a restaurant, you choose randomly between two of today’s specials, TS1 and TS2, on the menu. Let A denote{today’s special on first visit is TS1}, B denote { today’s special on second visit is TS1}, C denote { today’s special
5.62 Triple Blood Test Example 8 about the Triple Blood Test for Down syndrome found the results shown in the table on next column.a. Estimate the probability that Down syndrome occurs 1Down = Yes2.b. Find the estimated (i) sensitivity and (ii) specificity.c. Find the estimated (i) P1Yes •POS2
5.63 Simulating arrivals to local holiday center The director of a local holiday center is offering a special prize to the first married visitor. The distribution of the marital status for Americans estimated by SDA (Survey Documentation and Analysis) is shown below:The director decides that if
5.64 Probability of winning In Example 16, we estimated the probability of winning the game was 0.65.a. If you conducted 30 more simulations of this game, what probability of winning would you expect to get?b. The simulation in the example consisted of 20 repetitions.For a total of 200, conduct
5.80 Passing scores Suppose you need a score of 70 out of 100 to pass an exam. Let B denote the event you score at least 85 and D denote the event you pass the exam.Assume P1B2 = 0.3 and P1D2 = 0.7.a. Find P(B and D). When an event B is contained within an event D as here, explain why P(B and D) =
5.79 Car and phone You want to buy a car and a phone from a selection of 4 models and 6 models, respectively. Supposing all the models are equally likely to be chosen, denote the outcomes of your purchase by (Car, Phone), such as (3, 5)for the car labeled 3 and for the phone labeled 5.a. List the
5.78 Laundry detergent A manufacturer of laundry detergent has introduced a new product that it claims to be more environmentally sound. An extensive survey gives the percentages shown in the table.a. Estimate the probability that a randomly chosen consumer would have seen advertising for the new
5.77 Teens and parents In the CASA teen survey described in the previous exercise, 33% of teens reported that parents are never present at parties they attend. Thirtyone percent of teens who say parents are never present during parties report that marijuana is available at the parties they attend,
5.76 Teens and drugs In August 2006 the Center on Addiction and Substance Abuse (CASA) at Columbia University reported results of a survey of 1297 teenagers about their views on the use of illegal substances. Twenty percent of the teens surveyed reported going to clubs for music or dancing at least
5.75 Health insurance According to a 2006 census bureau report, 59% of Americans have private health insurance, 25%have government health insurance (Medicare or Medicaid or military health care), and 16% have no health insurance.a. Estimate the probability that a patient has health insurance.b.
5.74 Independent on coffee? Students in a geography class are asked whether they’ve visited Europe in the past 12 months and whether they’ve flown on a plane in the past 12 months.a. For a randomly selected student, would you expect these events to be independent or dependent? Explain.b. How
5.73 Driver’s exam Three 15-year-old friends with no particular background in driver’s education decide to take the written part of the Georgia Driver’s Exam. Each exam was graded as a pass (P) or a failure (F).a. How many outcomes are possible for the grades received by the three friends
5.72 Death penalty jury In arguing against the death penalty, Amnesty International has pointed out supposed inequities, such as the many times a black person has been given the death penalty by an all-white jury. If jurors are selected randomly from an adult population, find the probability that
5.71 Life after death In a General Social Survey, in response to the question “Do you believe in life after death?” 1455 answered yes and 332 answered no.a. Based on this survey, estimate the probability that a randomly selected adult in the United States believes in life after death.b. A
5.70 Caught doctoring the books After the major accounting scandals with Enron, a large energy company, the question may be posed, “Was there any way to examine Enron’s accounting books to determine if they had been doctored?”One way uses Benford’s law, which states that in a variety of
5.69 Choices for lunch For the set lunch at Amelia’s Restaurant, customers can select one meat dish, one vegetable, one beverage, and one dessert. The menu offers two meats (beef and chicken), three vegetables(corn, green beans, or potatoes), three beverages (cola, ice tea, or coffee), and one
5.68 P(life after death) Explain the difference between the relative frequency and subjective definitions of probability.Illustrate by explaining how it is possible to give a value for (a) the probability of life after death, (b) the probability that in the morning you remember at least one dream
5.67 Due for a boy? A couple has five children, all girls. They are expecting a sixth child. The father tells a friend that by the law of large numbers the chance of a boy is now much greater than 1/2. Comment on the father’s statement.
5.66 Peyton Manning completions As of the end of the 2010 NFL season, Indianapolis Colts quarterback Peyton Manning, throughout his 13-year career, completed 65%of all of his pass attempts. Suppose the probability each pass attempted in the next season has probability 0.65 of being completed.a.
5.65 Probability of winning In Example 16, we explored the number of rolls it takes to win the game. In reality, it’s not the number of rolls but rather the time it takes to move 12 spaces that dictates who wins the game. Consider two alternative strategies for playing the game. The first
5.45 Online sections For a course with two sections, let A denote {first section is online}, let B denote (at least one section is online}, and let C denote {both sections are online}.Suppose P (a section is online) = 1/2 and that the sections are independent.a. Find P 1C •A2 and P1C•B2.b. Are
5.44 Big loser in Lotto Example 10 showed that the probability of having the winning ticket in Lotto South was 0.00000007.Find the probability of holding a ticket that has zero winning numbers out of the 6 numbers selected (without replacement)for the winning ticket out of the 49 possible numbers.
5.43 Drawing more cards A standard deck of poker playing cards contains four suits (clubs, diamonds, hearts, and spades) and 13 different cards of each suit. During a hand of poker, 5 of the 52 cards have been exposed. Of the exposed cards, 3 were diamonds. Tony will have the opportunity to draw
5.23 Seat belt use and auto accidents Based on records of automobile accidents in a recent year, the Department of Highway Safety and Motor Vehicles in Florida reported the counts who survived (S) and died (D), according to whether they wore a seat belt 1Y = yes, N = no2. The data are presented in
5.22 Pick the incorrect statement Which of the following statements is not correct, and why?a. Last night, you randomly selected a restaurant for dinner from three similar restaurants in your city, with no prior preference for any one of them over the others. If your dining experience at the chosen
5.21 Insurance Every year the insurance industry spends considerable resources assessing risk probabilities. To accumulate a risk of about one in a million of death, you can drive 100 miles, take a cross country plane flight, work as a police officer for 10 hours, work in a coal mine for 12 hours,
5.20 Pick the incorrect statement Which of the following statements is not correct, and why?a. If the number of male and female employees at a call center is equal, then the probability you call four times and a female employee answers your call only once is 1/5.b. You have created a playlist of
5.19 Three children A couple plans to have three children.Suppose that the probability of any given child being female is 0.5, and suppose that the genders of each child are independent events.a. Write out all outcomes in the sample space for the genders of the three children.b. What should be the
5.18 On-time arrival probabilities The all-time, on-time arrival rate of a certain airline to a specific destination is 82%. This week, you have booked two flights to this destination with this airline.a. Construct a sample space for the on-time or late arrival of the two flights.b. Find the
5.17 Curling In the sport of curling, each shot is given points on a scale from 0–5, rating the success of each. Your friend claims, “Since the sum of points awarded when two shots are made is between 0 and 10, there is a one in eleven chance for each resulting sum to occur.” Do you agree or
5.16 More true-false questions Your teacher gives a true-false pop quiz with 10 questions.a. Show that the number of possible outcomes for the sample space of possible sequences of 10 answers is 1024.b. What is the complement of the event of getting at least one of the questions wrong?c. With
5.15 Pop quiz A teacher gives a four-question unannounced true-false pop quiz, with two possible answers to each question.a. Use a tree diagram to show the possible response patterns in terms of whether any given response is correct or incorrect.How many outcomes are in the sample space?b. An
5.14 Songs Out of 100 songs on a playlist, 15 are of your favorite artist. You decide to randomly play one track from this playlist.a. State the sample space for the possible outcomes.b. State the probability for each possible outcome.c. What is the probability that the track chosen randomly from
Student union poll Part of a student opinion poll at a university asks students what they think of the quality of the existing student union building on the campus.The possible responses were great, good, fair, and poor. Another part of the poll asked students how they feel about a proposed fee
Stock market randomness An interview in an investment magazine (In the Vanguard, Autumn 2003) asked mathematician John Allen Paulos, “What common errors do investors make?” He answered, “People tend not to believe that markets move in random ways. Randomness is difficult to recognize. If you
5.11 Unannounced pop quiz A teacher announces a pop quiz for which the student is completely unprepared. The quiz consists of 100 true-false questions. The student has no choice but to guess the answer randomly for all 100 questions.a. Simulate taking this quiz by random guessing. Number a sheet of
5.10 Simulate coin flips Use the web app Random Numbers(go to the tab that says Coin Flips) on the book’s website or other software (such as random.org/coin) to illustrate the long-run definition of probability by simulating shortterm and long-term results of flipping a balanced coin.a. Keep the
5.9 Nuclear war You are asked to use your best judgment to estimate the probability that there will be a nuclear war within the next 10 years. Is this an example of relative frequency or subjective definition of probability? Explain.
5.8 Heart transplant Before the first human heart transplant, Dr. Christiaan Barnard of South Africa was asked to assess the probability that the operation would be successful. Did he need to rely on the relative frequency definition or the subjective definition of probability? Explain.
Protecting the environment When the General Social Survey most recently asked subjects whether they are a member of an environmental group (variable GRNGROUP) and whether they would be willing to pay higher prices to protect the environment (variable GRNPRICE), the results were as shown in the
Global warming and trees A survey asks subjects whether they believe that global warming is happening (yes or no)and how much fuel they plan to use annually for automobile driving in the future, compared to their past use (less, about the same, more).a. Show the sample space of possible outcomes by
Newspaper sales You are the director of newspaper sales for the local paper. Each customer has signed up for either weekday delivery or weekend delivery. You record whether he or she received the delivery as Y for yes and N for no. The probabilities of the customer receiving the newspaper are as
5.42 Discussion with students In a statistics class of 30 students, 20 students are from the business program and 10 students are from the science program. The instructor randomly select three students, successively and without replacement, to discuss a question.a. True or false: The probability of
5.41 Answering homework questions Each question of an online homework consists of two parts. The probability that you answer the first part of a given question correctly is 0.75. Given that you answered the first part correctly, the probability you answer the second part correctly is 0.60.Given
5.40 Petra Kvitova serves Petra Kvitova of the Czech Republic won the 2014 Wimbledon Ladies’ Singles Championship. In the final game against Eugenie Bouchard of Canada she had 41 first serves, of which 28 were good, and three double faults.a. Find the probability that her first serve is good.b.
5.39 Happiness in relationship Are people happy in their romantic relationships? The table shows results from the 2012 General Social Survey for adults classified by gender and happiness.a. Estimate the probability that an adult is very happy in his or her romantic relationship.b. Estimate the
5.38 Obesity in America A 2014 Gallup poll reported that 27%of people in the United States are obese (having a body mass index score of 30 or more). Blacks have the highest obesity rate at 35%, whereas Asians have the lowest, at 9%.a. Of the three percentages (estimated probabilities)reported,
5.37 Down syndrome again Example 8 discussed the Triple Blood Test for Down syndrome, using data summarized in a table shown again below.a. Given that a test result is negative, show that the probability the fetus actually has Down syndrome is P1D•NEG2 = 0.0015.b. Is P1D•NEG2 equal to
5.36 Homeland security According to an article in The New Yorker (March 12, 2007), the Department of Homeland Security in the United States is experimenting with installing devices for detecting radiation at bridges, tunnels, roadways, and waterways leading into Manhattan. The New York Police
5.35 Identifying spam An article4 on www.networkworld.com about evaluating e-mail filters that are designed to detect spam described a test of MailFrontier’s Anti-Spam Gateway(ASG). In the test, there were 7840 spam messages, of which ASG caught 7005. Of the 7053 messages that ASG identified as
5.34 Go Celtics! Larry Bird, who played pro basketball for the Boston Celtics, was known for being a good shooter.In games during 1980–1982, when he missed his first free throw, 48 out of 53 times he made the second one, and when he made his first free throw, 251 out of 285 times he made the
5.33 Revisiting seat belts and auto accidents The following table is from Exercise 5.23 classifying auto accidents by survival status 1S = survived, D = died2 and seat belt status of the individual involved in the accident.a. Estimate the probability that the individual died (D) in the auto
5.32 Labor force In 2014, a sample of 1925 Americans revealed that about 20.5% of them belong to the government sector. 7.5% of these are part-time employees, 60%are full-time employees, and 32.5% are retired.a. Define events and identify which of these four probabilities refer to conditional
5.31 Religious affiliation The 2012 Statistical Abstract of the United States3 provides information on individuals’self-described religious affiliations. The information for 2008 is summarized in the following table (all numbers are in thousands).a. Find the probability that a randomly selected
5.30 Audit and low income Table 5.3 on audit status and income follows. Show how to find the probability of:a. Being audited, given that the taxpayer is in the lowest income category.b. Being in the lowest income category, given that the taxpayer is audited. Audited Income No Yes $1mil 0.0022 0.0003
5.29 Smoke alarms statistics National estimates of reported fires derived from the National Fire Incident Reporting System (NFIRS) and the National Fire Protection Association’s (NFPA’s) fire department survey show that in 2009–2013, 38% of home fire deaths occurred in homes with no smoke
Recidivism rates A 2014 article from Business Insider(http://www.businessinsider.com/department-of-justicereport-shows-high-recidivism-rate-2014-4) discusses recidivism rates in the United States. Recidivism is defined as being reincarcerated within five years of being sent to jail initially. Among
Arts and crafts sales A local downtown arts and crafts shop found from past observation that 20% of the people who enter the shop actually buy something. Three potential customers enter the shop.a. How many outcomes are possible for whether the clerk makes a sale to each customer? Construct a tree
5.7 Sample size and sampling accuracy Your friend is interested in estimating the proportion of people who would vote for his project in a local contest. He selects a large sample among his many friends and claims that, with such a large sample, he does not need to worry about the method of
6.33 SAT versus ACT SAT math scores follow a normal distribution with an approximate m = 500 and s = 100.Also ACT math scores follow a normal distrubution with an approximate m = 21 and s = 4.7. You are an admissions officer at a university and have room to admit one more student for the upcoming
5.76 Teens and drugs In August 2006 the Center on Addiction and Substance Abuse (CASA) at Columbia University reported results of a survey of 1297 teenagers about their views on the use of illegal substances. Twenty percent of the teens surveyed reported going to clubs for music or dancing at least
5.55 A true coincidence of emergency E-Comm, British Columbia’s emergency communications center, provides communication services and support systems to two million residents of southwest British Columbia, Canada. On any given day, the probability a randomly selected resident decides to call
5.54 Monkeys typing Shakespeare Since events of low probability eventually happen if you observe enough trials, a monkey randomly pecking on a typewriter could eventually write a Shakespeare play just by chance. Let’s see how hard it would be to type the title of Macbeth properly.Assume 50 keys
5.53 Coincidence in your life State an event that has happened to you or to someone you know that seems highly coincidental (such as seeing a friend while on vacation).Explain why that event may not be especially surprising, once you think of all the similar types of events that could have happened
5.52 Horrible 11 on 9/11 The digits in 9/11 add up to 11 19 + 1 + 12, American Airlines flight 11 was the first to hit the World Trade Towers (which took the form of the number 11), there were 92 people on board 19 + 2 = 112, September 11 is the 254th day of the year 12 + 5 + 4 = 112, and there are
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