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Statistics The Art And Science Of Learning From Data 4th Global Edition Alan Agresti, Christine A. Franklin, Bernhard Klingenberg - Solutions
6.17 Probability in graph For the normal distributions shown below, use Table A, software, or a calculator to find the probability that an observation falls in the shaded region. a. P b. +0.670 -0.50 +0.5
6.16 Probabilities in tails For a normal distribution, use Table A, software, or a calculator to find the probability that an observation isa. at least 1 standard deviation above the mean.b. at least 1 standard deviation below the mean.c. within 1 standard deviation of the mean.For each part,
5.115 Generalizing the multiplication rule For events A, B, and C, explain why P1A and B and C2 =P1A2 * P1B•A2 * P1C•A and B2.
5.114 Generalizing the addition rule For events A, B, and C such that each pair of events is disjoint, use a Venn diagram to explain why P1A or B or C2 = P1A2 + P1B2 + P1C2.
5.113 Prosecutor’s fallacy An eyewitness to the crime says that the person who committed it was male, between 15 and 20 years old, Hispanic, and drove a blue Honda. The prosecutor points out that a proportion of only 0.001 people living in that city match all those characteristics, and one of
5.92 Prostate cancer A study of the PSA blood test for diagnosing prostate cancer in men (by R. M. Hoffman et al., BMC Family Practice, vol. 3, p. 19, 2002) used a sample of 2620 men who were 40 years and older. When a positive diagnostic test result was defined as a PSA reading of at least 4, the
5.91 HIV testing For a combined ELISA-Western blot blood test for HIV positive status, the sensitivity is about 0.999 and the specificity is about 0.9999 (Gigerenzer 2002, pp. 124, 126).a. Consider a high-risk group in which 10% are truly HIV positive. Construct a tree diagram to summarize this
5.90 Color blindness For genetic reasons, color blindness is more common in men than women: 5 in 100 men and 25 in 10,000 women suffer from color blindness.a. Define events and identify in words these proportions as conditional probabilities.b. If the population is half male and half female, what
5.89 Screening for colorectal cancer Gigerenzer (2002, p. 105) reported that on the average, “Thirty out of every 10,000 people have colorectal cancer. Of these 30 people with colorectal cancer, 15 will have a positive hemoccult test. Of the remaining 9,970 people without colorectal cancer, 300
5.88 Screening for heart attacks Biochemical markers are used by emergency room physicians to aid in diagnosing patients who have suffered acute myocardial infarction(AMI), or what’s commonly referred to as a heart attack.One type of biochemical marker used is creatine kinase (CK). Based on a
5.87 Screening smokers for lung cancer An article about using a diagnostic test (helical computed tomography) to screen adult smokers for lung cancer warned that a negative test may cause harm by providing smokers with false reassurance, and a false-positive test results in an unnecessary operation
5.86 Death penalty and false positives For the decision about whether to convict someone charged with murder and give the death penalty, consider the variables reality (defendant innocent, defendant guilty) and decision (convict, acquit).a. Explain what the two types of errors are in this
5.85 Amazing roulette run? A roulette wheel in Monte Carlo has 18 even-numbered slots, 18 odd-numbered slots, a slot numbered zero, and a double zero slot. On August 18, 1913, it came up even 26 times in a row.8 As more and more evens occurred, the proportion of people betting on an odd outcome
5.84 No coincidences Over time, you have many conversations with a friend about your favorite actress, favorite musician, favorite book, favorite TV show, and so forth for 100 topics. On any given topic, there’s only a 0.02 probability that you agree. If you did agree on a topic, you would
5.83 A dice game Consider a game in which you roll two dice, and you win if the total is 7 or 11 and you lose if the total is 2, 3, or 12. You keep rolling until one of these totals occurs. Using conditional probability, find the probability that you win.
5.82 Waste dump sites A federal agency is deciding which of two waste dump projects to investigate. A top administrator estimates that the probability of federal law violations is 0.30 at the first project and 0.25 at the second project. Also, he believes the occurrences of violations in these two
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.
5.80 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
5.94 Win again Exercise 5.65 discussed how to use simulation and the table of random digits to estimate an expected value. Referring to the previous exercise, conduct a simulation consisting of at least 20 repetitions to estimate the expected number of combo meals one would need to purchase to win
5.95 Simulate law of large numbers Using the Random Numbers web app (or other software), simulate the flipping of balanced and biased coins.a. Report the proportion of heads after (i) 10 flips,(ii) 100 flips, (iii) 1000 flips, and (iv) 10,000 flips.(Select the Coin Flips tab in the app. You can
5.96 Illustrate probability terms with scenariosa. What is a sample space? Give an example of a sample space for a scenario involving (i) a designed experiment and (ii) an observational study.b. What are disjoint events? Give an example of two events that are disjoint.c. What is a conditional
5.112 Driving versus flying In the United States in 2002, about 43,000 people died in auto crashes and 0 people died in commercial airline accidents. G. Gigerenzer (2002, p. 31)states, “The terrorist attack on September 11, 2001, cost the lives of some 3,000 people. The subsequent decision of
5.111 Obesity in Australia A report published in 2016 (www.huffingtonpost.com.au) showed that 63.4% of Australian adults were overweight or obese in 2014–2015. When you randomly select two adults from the entire population for estimating the percentage of obese Australians, what is wrong in the
5.110 True or false Answer true of false for each part.a. When you flip a coin ten times, you are more likely to get the sequence HHHHHTTTTT than the sequence HHHHHHHHHH.b. When you flip a coin ten times, you are more likely to get a sequence that contains five heads than a sequence that contains
5.109 Multiple choice: Comparable risks Mammography is estimated to save about 1 life in every 1000 women.“Participating in annual mammography screening … has roughly the same effect on life expectancy as reducing the distance one drives each year by 300 miles” (Gigerenzer 2002, pp. 60, 73).
5.108 Multiple choice: Dream come true You have a dream in which you see your favorite movie star in person. The very next day, you are visiting Manhattan and you see her walking down Fifth Avenue.a. This is such an incredibly unlikely coincidence that you should report it to your local
5.107 Multiple choice: Coin flip A balanced coin is flipped 100 times. By the law of large numbers:a. There will almost certainly be exactly 50 heads and 50 tails.b. If we got 100 heads in a row, almost certainly the next flip will be a tail.c. For the 100 flips, the probability of getting 100
5.106 Multiple choice Which of the following is always true?a. If A and B are independent, then they are also disjoint.b. P1A•B2 + P1A•Bc2 = 1c. If P1A•B2 = P1B•A2, then A and B are independent.d. If A and B are disjoint, then A and B cannot occur at the same time.
5.105 Multiple choice Choose ALL correct responses. For two events A and B, P1A2 = 0.5 and P1B2 = 0.2. Then P(A or B) equalsa. 0.10, if A and B are independentb. 0.70, if A and B are independentc. 0.60, if A and B are independentd. 0.70, if A and B are disjoint
5.104 Streak shooter Sportscaster Maria Coselli claims that players on the New York Knicks professional basketball team are streak shooters. To make her case, she looks at the statistics for all the team’s players over the past three games and points out that one of them (Joe Smith)made six shots
5.103 Protective bomb Before the days of high security at airports, there was a legendary person who was afraid of traveling by plane because someone on the plane might have a bomb. He always brought a bomb himself on any plane flight he took, believing that the chance would be astronomically small
5.102 How good is a probability estimate? In Example 8 about Down syndrome, we estimated the probability of a positive test result (predicting that Down syndrome is present) to be P1POS2 = 0.257, based on observing 1355 positive results in 5282 observations. How good is such an estimate?From
5.101 FIFA World Cup In the FIFA World Cup, 32 teams play 64 matches. Of these, 50% qualify for the round of 16 series of matches. Half of the teams in the round of 16 continue to the quarterfinals, and of these, 50% continue to the semifinals. The winners of the semifinals compete in the final
5.100 Marijuana leads to heroin? Nearly all heroin addicts have used marijuana sometime in their lives. So, some argue that marijuana should be illegal because marijuana users are likely to become heroin addicts. Use a Venn diagram to illustrate the fallacy of this argument by sketching sets for M
5.99 Mrs. Test Mrs. Test (see www.mrstest.com) sells diagnostic tests for various conditions. Its website gives only imprecise information about the accuracy of the tests. The test for pregnancy is said to be “over 99% accurate.” Describe at least four probabilities to which this could refer.
5.98 Risk of space shuttle After the Columbia space shuttle disaster, a former NASA official who faulted the way the agency dealt with safety risk warned (in an AP story, March 7, 2003) that NASA workers believed, “If I’ve flown 20 times, the risk is less than if I’ve flown just once.”a.
5.97 Short term versus long run According to countrymeters.info, the American population in 2016 consisted of 49.4%males and 50.6% females. To illustrate how short-term aberrations do not affect the long run, suppose that you select 10 Americans and you get 10 males.a. Find the cumulative
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,
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.
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
6.55 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
6.56 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
6.57 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
6.58 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.
6.59 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.
6.60 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.
6.61 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 +
6.62 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
6.64 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
6.65 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
6.66 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
6.67 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
6.68 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
6.52 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
6.51 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
6.35 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
6.36 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
6.37 Symmetric binomial Construct a graph similar to that in Figure 6.1 for each of the following binomial distributions:a. n = 4 and p = 0.50.b. n = 4 and p = 0.30.c. n = 4 and p = 0.10.d. Which if any of the graphs in parts a–c are symmetric?Without actually constructing the graph, would the
6.38 Unfair wealth distribution sentiment According to a study published in www.gallup.com in 2015, 63% of Americans said wealth should be more evenly distributed among a larger percentage of people. For a sample of 10 Americans, let X = number of respondents who said wealth was unfairly
6.39 Bidding on eBay You are bidding on four items available on eBay. You think that for each bid, you have a 25%chance of winning it, and the outcomes of the four bids are independent events. Let X denote the number of winning bids out of the four items you bid on.a. Explain why the distribution
6.40 More eBay bidding For each of the following situations, explain whether the binomial distribution applies for X.a. You are bidding on four items available on eBay. You think that you will win the first bid with probability 25%and the second through fourth bids with probability 30%.Let X denote
6.42 NBA shooting In the National Basketball Association, the top free throw shooters usually have probability of about 0.90 of making any given free throw.a. During a game, one such player (Dirk Nowitzki) shot 10 free throws. Let X = number of free throws made.What must you assume for X to have a
6.43 Season performance Refer to the previous exercise. Over the course of a season, this player shoots 400 free throws.a. Find the mean and standard deviation of the probability distribution of the number of free throws he makes.b. By the normal distribution approximation, within what range would
6.44 Is the die balanced? A balanced die with six sides is rolled 60 times.a. For the binomial distribution of X = number of 6s, what is n and what is p?b. Find the mean and the standard deviation of the distribution of X. Interpret.c. If you observe x = 0, would you be skeptical that the die is
6.45 Exit poll An exit poll is taken of 3000 voters in a statewide election. Let X denote the number who voted in favor of a special proposition designed to lower property taxes and raise the sales tax. Suppose that in the population, exactly 50% voted for it.a. Explain why this scenario would seem
6.47 Poor, poor, Pirates On September 7, 2008, the Pittsburgh Pirates lost their 82nd game of the 2008 season and tied the 1933–1948 Philadelphia Phillies major sport record (baseball, football, basketball, and hockey)for most consecutive losing seasons at 16. In fact, their losing streak
6.48 Checking guidelines In a village having more than 100 adults, eight are randomly selected in order to form a committee of residents. 40% of adults in the village are connected with agriculture.a. Verify that the guidelines have been satisfied about the relative sizes of the population and the
6.49 Movies sample Five of the 20 movies running in movie theatres this week are comedies. A selection of four movies are picked at random. Does X = the number of movies in the sample which are comedies have the binomial distribution with n = 4 and p = 0.25? Explain why or why not.
6.50 Binomial needs fixed n For the binomial distribution, the number of trials n is a fixed number. Let X denote the number of girls in a randomly selected family in Canada that has three children. Let Y denote the number of girls in a randomly selected family in Canada (that is, the number of
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
Fast-food profits Mac’s fast-food restaurant finds that its daily profits have a normal distribution with mean$140 and standard deviation $80.a. Find the probability that the restaurant loses money on a given day (that is, daily profit less than 0).b. Find the probability that the restaurant
Babies in China The sex distribution of new babies is close to 50% each, with the percentage of males usually being just slightly higher. In China in recent years, the percentage of female births seems to have dropped, a combination of policy limiting family size, the desire of most families to
6.90 TRUE or FALSE? IQR for normal distribution For a normally distributed random variable, the IQR is larger than the length of the interval m { s.
6.91 Multiple choice: Guess answers A question has four possible answers, only one of which is correct. You randomly guess the correct response. With 20 such questions, the distribution of the number of incorrect answersa. is binomial with n = 20 and p = 0.25.b. is binomial with n = 20 and p =
6.92 Multiple choice: Terrorist coincidence? On 9/11/2002, the first anniversary of the terrorist destruction of the World Trade Center in New York City, the winning three-digit New York State Lottery number came up 9-1-1. The probability of this happening wasa. 1>1000.b. 11>100022 = 0.000001.c. 1
6.93 SAT and ethnic groups Lake Wobegon Junior College admits students only if they score above 1200 on the sum of their critical reading, mathematics, and writing scores. Applicants from ethnic group A have a mean of 1500 and a standard deviation of 300 on this test, and applicants from ethnic
6.94 College acceptance The National Center for Educational Statistics reported that in 2012 the ACT college placement and admission examination had a mean of 21.1 and standard deviation of 5.3. (Source: Data from nces.ed.gov/programs/digest/d12/tables/dt12_147.asp.)a. Which probability
6.95 Standard deviation of a discrete probability distribution The variance of a probability distribution of a random variable is a weighted average of its squared distances from the mean m. For discrete random variables, it equals s2 = 1x - m22P1x2.Multiply each possible squared deviation 1x - m22
Mean and standard deviation for a binary random variable The previous exercise gave the formula for the standard deviation of a discrete random variable X. Let’s look at a simple case. Suppose X is a binary random variable where X = 1 with probability p and X = 0 with probability 11 - p2. a. Show
Linear transformations: Taxes and fees Assume prices for used Audi A4s available on carmax.com follow a normal distribution with mean $23,800 and standard deviation $4,380.a. Prices quoted on carmax.com are without sales tax.To account for a 6% sales tax, you multiply each price by 1.06. What is
Binomial probabilities Justify the px11 - p2n - x part of the binomial formula for the probability P(x) of a particular sequence with x successes, using what you learned in Section 5.2 about probabilities for intersections of independent events.
Waiting time for doubles Most discrete random variables can take on a finite number of values. Let X = the number of rolls of two dice necessary until doubles (the same number on each die) first appears.The possible values for this discrete random variable(called the geometric) are 1, 2, 3, 4, 5,
World Series in baseball For the 68 World Series played between 1945 and 2013, 13 (19%) ended after 4 games, 12 (18%) after 5 games, 15 (22%) after 6 games, and 28(41%) needed to go all the way to 7 games.a. Find the average number of games played based on these data. (Remember from Section 6.1
Best of seven games In professional baseball, basketball, and hockey in North America, the final two teams in the playoffs play a best of seven series of games. The first team to win four games is the champion. Use simulation with the Random Numbers app on the text’s Web site to approximate the
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
6.2 Dental Insurance You plan to purchase dental insurance for your three remaining years in school. The insurance makes a one-time payment of $1,000 in case of a major dental repair (such as an implant) or $100 in case of a minor repair (such as a cavity). If you don’t need dental repair over
Airline overbooking For the Boston to Chicago route, an airline flies a Boeing 737–800 with 170 seats. Based on past experience, the airline finds that people who purchase a ticket for this flight have 0.80 probability of showing up for the flight. They routinely sell 190 tickets for the flight,
Stock market randomness Based on the previous exercise and what you have learned in this and the previous chapter (for example, Exercise 5.12), if you are a serious investor, explain why you should not get too excited if sometime in the next year the stock market goes up for seven days in a row.
Metric height A Dutch researcher reads that male height in the Netherlands has a normal distribution with m = 72.0 inches and s = 4.0 inches. She prefers to convert this to the metric scale 11 inch = 2.54 centimeters2.The mean and standard deviation then have the same conversion factor.a. In
Manufacturing tennis balls According to the rules of tennis, a tennis ball is supposed to weigh between 56.7 grams (2 ounces) and 58.5 grams (2 1>16 ounces).A machine for manufacturing tennis balls produces balls with a mean of 57.6 grams and a standard deviation of 0.3 grams when it is operating
Bride’s choice of surname According to a study done by the Lucy Stone League and reported by ABC News 9 in February 2011, 90% of brides take the surname of their new husband. Ann notes that of her four best friends who recently married, none kept her own name. If they had been a random sample of
More ESP In another experiment with Jane Doe from the previous exercise, she had to predict which of five numbers was chosen in each of three trials. Jane Doe does not actually have ESP and is just randomly guessing.Explain why this experiment satisfies the three conditions for the binomial
Weather A weather forecaster states, “The chance of rain is 50% on Saturday and 50% again on Sunday. So there’s a 100% chance of rain sometime over the weekend.”If whether it rains on Saturday is independent of whether it rains on Sunday, find the actual probability of rain at least once
Dating success Based on past experience, Julio believes he has a 60% chance of success when he calls a woman and asks for a date.a. State assumptions needed for the binomial distribution to apply to the number of times he is successful on his next five requests.b. If he asks the same woman each of
Total loss A total-loss insurance policy pays $15,000 against a premium of $500 if a car is considered a total loss. A company estimates that for such policies, the probability a car is considered a total loss is 0.0015. In a portfolio of n policies of this type, let X be the number of total loss
Likes on Facebook A large retail chain sends out an ad about a new product to 15 million users on Facebook and asks them to like the message. Assume the probability that a user will like the message (instead of ignoring it) is 0.00001.a. State assumptions for a binomial distribution to apply for X
Survival A cohort life table is used to represent the overall mortality rate of the entire lifetime of a certain population.Assume the number of people who survived until the age of 1 in a certain cohort is 98500 with a probability of death of 0.0002 between the ages of 1 and 4.a. Find the expected
6.82 Which distribution for sales? A salesperson uses random digit dialing to call people and try to interest them in applying for a charge card for a large department store chain. From past experience, she is successful on 2% of her calls. In a typical working day, she makes 200 calls.Let X be the
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