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Statistics The Art And Science Of Learning From Data 4th Global Edition Alan Agresti, Christine A. Franklin, Bernhard Klingenberg - Solutions
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
5.45 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
5.42 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.
5.56 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
5.57 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
5.58 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
5.74 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
5.64 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
5.63 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
5.62 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
5.61 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
5.39 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
5.17 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.
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
5.6 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 that a call is answered by a female employee is 0.50.b. If you randomly generate 10 digits, each integer between 0
5.4 Airline accident deaths Airplane safety has been improving over the years. From 2000 to 2010, the average number of global airline deaths per year was over 1000, even when excluding the nearly 3000 deaths in the United States on September 11, 2001. The number of global airline deaths declined
5.3 Counselor availability You visit your counselor’s office at 10 randomly chosen times, and he is not available at any of those times. Does this mean that the probability of your counselor being available at his office for students equals 0? Explain.
5.2 Minesweeper The objective of the game Minesweeper is to clear a field without detonating any mines or bombs.Your friend claims that his rate of completing the game successfully is 90%.a. You decide to challenge your friend. He makes 10 attempts to complete the game, but is successful in only 7
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
5.19 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
5.20 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
5.36 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
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
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
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
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,
5.1 Probability Explain what is meant by the long-run relative frequency definition of probability.
6.32 Automatic filling machine A machine is programmed to fill packets with 500 grams of nuts. It is known from previous experiences the net weight of nuts in the packets are normally distributed with m = 502 grams and s = 3 grams. A packet is considered conformant with the weight specifications if
6.12 Buying on eBay You are watching two items posted for sale on eBay and bid $30 for the first and $20 for the second item. You estimate that you are going to win the first bid with probability 0.1 and the second bid with probability 0.2, and you assume that winning the two bids are independent
6.11 Profit and the weather From past experience, a wheat farmer living in Manitoba, Canada, finds that his annual profit (in Canadian dollars) is $80,000 if the summer weather is typical, $50,000 if the weather is unusually dry, and $20,000 if there is a severe storm that destroys much of his
6.10 Ideal number of children Let X denote the response of a randomly selected person to the question, “What is the ideal number of children for a family to have?” The probability distribution of X in the United States is approximately as shown in the table, according to the gender of the
6.9 More Roulette The previous exercise on roulette described two bets: one bet on the single number 23 with winnings of either $350 or - $10 and a different bet on black with winnings of either $10 or - $10. For both types of bets, the expected winning is - $0.53. Which of the two bets has the
6.7 Playing the lottery The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999. The lottery selects a three-digit number at random.With a bet of $1, you win $500 if your number is selected and nothing
6.6 Selling houses Let X represent the number of homes a real estate agent sells during a given month. Based on previous sales records, she estimates that P102 = 0.68, P112 = 0.19, P122 = 0.09, P132 = 0.03, P142 = 0.01, with negligible probability for higher values of x.a. Explain why it does not
6.5 WhatsApp reviews 71% of WhatsApp users have given it a five-star rating on Google Play. Of the remaining users, 15%, 6%, 3%, and 5% have given ratings of four, three, two, and one stars, respectively to the application.a. Specify the probability distribution for the number of stars as rated by
6.4 Basketball shots To win a basketball game, two competitors play three rounds of one three-point shot each. The series ends if one of them scores in a round but the other misses his shot or if both get the same result in each of the three rounds. Assume competitors A and B have 30% and 20% of
6.3 San Francisco Giants hitting The table shows the probability distribution of the number of bases for a randomly selected time at bat for a San Francisco Giants player in 2010 (excluding times when the player got on base because of a walk or being hit by a pitch). In 74.29% of the at-bats the
6.1 Rolling dicea. State in a table the probability distribution for the outcome of rolling a balanced die. (This is called the uniform distribution on the integers 1, 2,c, 6.)b. Two balanced dice are rolled. Show that the probability distribution for X = total on the two dice is as shown in the
5.120 Saving a business The business you started last year has only $5000 left in capital. A week from now, you need to repay a $10,000 loan or you will go bankrupt. You see two possible ways to raise the money you need. You can ask a large company to invest the $10,000, wooing it with the $5000
5.119 Which tennis strategy is better? A tennis match can consist of the best of three sets (that is, the winner is the first to win two sets) or the best of five sets (the winner is the first to win three sets). Which would you be better off playing if you are the weaker player and have
5.118 Simulate table tennis In a table tennis game, the first person to get at least 11 points while being ahead of the opponent by at least two points wins the game. In games between you and an opponent, suppose successive points are independent and that the probability of your winning any given
5.117 Simulating matching birthdays Do you find it hard to believe that the probability of at least one birthday match in a class of 25 students is 0.57? Let’s simulate the answer.Using the Random Numbers web app accessible from the book’s website, each student in the class should simulate 25
5.116 Bayes’ rule Suppose we know P1A2, P1B•A2, and P1Bc •Ac2, but we want to find P1A•B2. a. Using the definition of conditional probability for P(A|B) and for P(B|A), explain why P(A|B) - P(A and B)/P(B) [P(A)P(B|A)]/P(B). b. Splitting the event that B occurs into two parts, according to
6.13 Selling at the right price An insurance company wants to examine the views of its clients about the prices of three car insurance plans launched last year. It conducts a survey with two sets of plans with different prices and finds that:• If plan A is sold for $150, plan B for $250, and plan
6.14 Uniform distribution A random number generator is used to generate a real number between 0 and 1, equally likely to fall anywhere in this interval of values. (For instance, 0.3794259832c is a possible outcome.)a. Sketch a curve of the probability distribution of this random variable, which is
6.15 TV watching A social scientist uses the General Social Survey (GSS) to study how much time per day people spend watching TV. The variable denoted by TVHOURS at the GSS Web site measures this using the discrete values 0, 1, 2, c , 24.a. Explain how, in theory, TV watching is a continuous random
6.31 April precipitation Over roughly the past 100 years, the mean monthly April precipitation in Williamstown, Massachusetts, equaled 3.6 inches with a standard deviation of 1.6 inches. (Source: http://web.williams.edu/weather/)a. In April 1983, the wettest April on record, the precipitation
6.30 Quartiles and outliers For an approximately normally distributed random variable X with a mean of 200 and a standard deviation of 36,a. Find the z-score corresponding to the lower quartile and upper quartile of the standard normal distribution.b. Find and interpret the lower quartile and upper
6.28 Birth weight for boys In the United States, the mean birth weight for boys is 3.41 kg, with a standard deviation of 0.55 kg. (Source: cdc.com.) Assuming that the distribution of birth weight is approximately normal, find the following using a table, calculator, or software.a. A baby is
6.27 Lifespan of phone batteries Most phones use lithium-ion(Li-ion) batteries. These batteries have a limited number of charge and discharge cycles, usually falling between 300 and 500. Beyond this lifespan, a battery gradually diminishes below 50% of its original capacity.a. Suppose the
6.26 Coffee Machine Suppose your favorite coffee machine offers 12 ounce cups of coffee. The actual amount of coffee put in the cup by the machine varies according to a normal distribution, with mean equal to 13 ounces and standard deviation equal to 0.6 ounces. For each question below, sketch a
6.25 Blood pressure A World Health Organization study(the MONICA project) of health in various countries reported that in Canada, systolic blood pressure readings have a mean of 121 and a standard deviation of 16. A reading above 140 is considered high blood pressure.a. What is the z-score for a
6.24 U.S. Air Force To join the U.S. Air Force as an officer, you cannot be younger than 18 or older than 34 years of age. The distribution of age of Americans in 2012 was normal with m = 38 years and s = 22.67 years. What proportion of U.S. citizens are not eligible to serve as an officer due to
6.23 z-score and central probability Find the z-score such that the interval within z standard deviations of the mean(between m - zs and m + zs) for a normal distribution containsa. 50% of the probability.b. 90% of the probability.c. Sketch the two cases on a single graph.
6.22 z-score for right-tail probabilitya. For the normal distribution shown below, find the z-score.b. Find the value of z (rounding to two decimal places)for right-tail probabilities of (i) 0.05 and (ii) 0.005. 02+11 .20
6.21 Probability in tails for given z-score For a normal distribution,a. Show that a total probability of 0.01 falls more than z = 2.58 standard deviations from the mean.b. Find the z-score for which the two-tail probability that falls more than that many standard deviations from the mean in either
6.20 z-score for given probability in tails For a normal distribution,a. Find the z-score for which a total probability of 0.04 falls more than z standard deviations (in either direction)from the mean, that is, below m - zs or above m + zs.b. For this z-score, explain why the probability of values
6.19 Central probabilities For a normal distribution, use Table A to verify that the probability (rounded to two decimal places) withina. 1.64 standard deviations of the mean equals 0.90.b. 2.58 standard deviations of the mean equals 0.99.c. Find the probability that falls within 0.67 standard
6.18 Empirical rule Verify the empirical rule by using Table A, software, or a calculator to show that for a normal distribution, the probability (rounded to two decimal places) withina. 1 standard deviation of the mean equals 0.68.b. 2 standard deviations of the mean equals 0.95.c. 3 standard
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