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Statistics The Art And Science Of Learning From Data 3rd Edition Alan Agresti, Christine A. Franklin - Solutions

A Reuters story (April 2, 2003) reported that “The number of heart attack victims fell by almost 60% at one hospital six months after a smoke-free ordinance went into effect in the area (Helena, Montana), a study showed, reinforcing concerns about second-hand smoke.” The number of hospital
A randomized clinical trial is planned for AIDS patients to investigate whether a new treatment provides improved survival over the current standard treatment. It is not known whether it will be better or worse. a. Why do researchers use randomization in such experiments, rather than letting the
Example 9 and Table 4.3 described a case-control study on smoking and lung cancer. Explain carefully why it is not sensible to use the study’s proportion of smokers who had lung cancer (that is, 688/(688 + 650) ) and proportion of nonsmokers who had lung cancer (21/(21 + 59)) to estimate
A vaccine is claimed to be effective in preventing a rare disease that occurs in about one of every 100,000 people. Explain why a randomized clinical trial comparing 100 people who get the vaccine to 100 people who do not get it is unlikely to be worth doing. Explain how you could use a
Exercise 1.10 discussed a study at Yale University’s Infant Cognition Center. Researchers were interested in determining whether infants had the ability to distinguish between the actions of helping and hindering. Each infant in the study was shown two videos. One video included a figure
In the previous exercise, we considered how showing each baby in the study the two videos in the same order might create a bias. In fact, of the 16 babies in the study, half were shown the videos in one order while the other half was shown the videos in the opposite order. Explain how to use the
Use the Simulating the Probability of Head with a Fair Coin applet on the text CD or other software to illustrate the long-run definition of probability by simulating short-term and long-term results of flipping a balanced coin. a. Set the sample size to n = 10. Run the applet 10 times, and record
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 = marijuana use and H = heroin
Suppose 80% of students finish high school. Of them, 50% finish college. Of them, 20% get a masters’ degree. Of them, 30% get a Ph.D. a. What percentage of students get a Ph.D.? b. Explain how your reasoning in part a used multiplications rule with conditional probabilities. c. Given that a
In Example 8 about Down syndrome, we estimated the probability of a positive test result (predicting that Down syndrome is present) to be P(POS) = 0.257, based on observing 1355 positive results in 5282 observations. How good is such an estimate? From Section 4.2, 1/1n is an approximate margin of
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 that two people on
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 in a row at one
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 paper 1 to 100 to
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 ten heads.
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 them is the defendant. Thus, the
For events A, B, and C such that each pair of events is disjoint, use a Venn diagram to explain why P(A or B or C) = P(A) + P(B) + P(C).
For events A, B, and C, explain why P(A and B and C) = P(A) × P(B |A) × P(C | A and B).
Suppose we know P(A),P(B | A), and P(Bc | Ac), but we want to find P(A | B). 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 whether or
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 applet on the text CD, each student in the class should simulate 25 numbers between 1 and 365. Observe whether there is at
In table tennis, the first person to get at least 21 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 suppose the probability of your winning any given point is 0.40.a. Do you have any
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 probability 0.40 of winning any particular set?a.
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 have people write down
The business you started last year as 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 them with the $5000 you have left. You guess
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 increase to help fund
A single random digit is selected using software or a random number table. a. State the sample space for the possible outcomes. b. State the probability for each possible outcome, based on what you know about the way random numbers are generated. c. Each outcome in a sample space must have
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 unprepared
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 random guessing, show that the
A couple plans to have two children. Each child is equally likely to be a girl or boy, with gender independent of that of the other child. a. Construct a sample space for the genders of the two children. b. Find the probability that both children are girls. c. Answer part b if in reality, for a
A couple plans on having three children. Suppose that the probability of any given child being female is 0.5, and also 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
Your friend decides to flip a coin repeatedly to analyze whether the probability of a head on each flip is 1/2. He flips the coin 10 times and observes a head 7 times. He concludes that the probability of a head for this coin is 7/10 = 0.70. a. Your friend claims that the coin is not balanced,
A couple plans on having four children. The father notes that the sample space for the number of girls the couple can have is 0, 1, 2, 3, and 4. He goes on to say that since there are five outcomes in the sample space, and since each child is equally likely to be a boy or girl, all five outcomes
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 (Y = yes, N = no). The data are presented in the contingency table shown.a. What is
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 very willing to pay higher prices to protect the environment (variable GRNPRICE), the results were as shown in the table. For a randomly
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 drawing a tree diagram
You are the marketing director for a museum that raises money by selling gift items from a mail-order catalog. For each catalog sent to a potential customer, the customer€™s entry in the data file is Y if they ordered something and N if they did not (Y = yes, N = no). After you have
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 diagram to show the
Because of the increasing nuisance of spam e-mail messages, many start-up companies have emerged to develop e-mail filters. One such filter was recently advertised as being 95% accurate. The way the advertisement is worded, 95% accurate could mean that (a) 95% of spam is blocked, (b) 95% of valid
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.
The 2011 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). Christian Catholic.................
Current estimates are that about 25% of all deaths are due to cancer, and of the deaths that are due to cancer, 30% are attributed to tobacco, 40% to diet, and 30% to other causes. a. Define events, and identify which of these four probabilities refer to conditional probabilities. b. Find the
The following table is from Exercise 5.23 classifying auto accidents by survival status (S = survived, D = died) and seat belt status of the individual involved in the accident.a. Estimate the probability that the individual died (D) in the auto accident. b. Estimate the probability that the
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 second one.a. Form a
An article 4 in www.networkworld.com about evaluating e-mail filters that are designed to detect spam described a test of Mail Frontier’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 spam, they were
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 Department (NYPD) has
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 P(D | NEG) = 0.0015. b. Is P(D | NEG) equal to P(NEG | D)? If so, explain why.
Workers specified as actively disengaged are those who are emotionally disconnected from their work and workplace. A Gallup poll conducted in December 2010 5 surveyed individuals who were either unemployed or who were actively disengaged in their current position. Individuals were asked to classify
Are people happy in their marriages? The table shows results from the 2008 General Social Survey for married adults classified by gender and level of happiness.a. Estimate the probability that a married adult is very happy. b. Estimate the probability that a married adult is very happy, (i) given
For the 10-year period between 2000 and 2010, the average number of deaths due to accidents involving U.S. commercial airline carriers has been about 46 per year. Over that same period, the average number of passengers has been more than 600 million per year. a. Can you consider this a long run or
Serena Williams won the 2010 Wimbledon Ladies’ Singles Championship. For the seven matches she played in the tournament, her total number of first serves was 379, total number of good first serves was 256, and total number of double faults was 15. a. Find the probability that her first serve is
Pro basketball player Shaquille O’Neal is a poor free-throw shooter. Consider situations in which he shoots a pair of free throws. The probability that he makes the first free throw is 0.50. Given that he makes the first, suppose the probability that he makes the second is 0.60. Given that he
A standard card deck has 52 cards consisting of 26 black and 26 red cards. Three cards are dealt from a shuffled deck, without replacement. a. True or false: The probability of being dealt three black cards is (1/2) × (1/2) × (1/2) = 1/8. If true, explain why. If false, show how to get the
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.
For a family with two children, let A denote {first child is female}, let B denote (at least one child is female}, and let C denote {both children are female}. a. Show that P(C | A) = 1/2. b. Are A and C independent events? Why or why not? c. Find P(C | B). d. Describe what makes P(C | A) different
In three independent flips of a balanced coin, let A denote {first flip is a head}, B denote {second flip is a head}, C denote {first two flips are heads}, and D denote {three heads on the three flips}. a. Find the probabilities of A, B, C, and D. b. Which, if any, pairs of these events are
Of the first 44 presidents of the United States (George Washington through Barack Obama), two had the same birthday (Polk and Harding). Is this highly coincidental? Answer by finding the probability of at least one birthday match among 44 people.
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 match for at least two
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. (You can pair 24 other students with each student, but how can you make sure you don’t count each pair twice?)
Each year, the Website espn.go.com/nba/hollinger/playoffodds displays the probabilities of professional basketball teams achieving certain goals. For example, at the end of the 2009€“2010 regular season, the site listed the following probabilities (expressed as percentages) of each of the
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 the par 3s are each short
The digits in 9/11 add up to 11 (9 + 1 + 1), 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 (9 + 2 = 11), September 11 is the 254th day of the year (2 + 5 + 4 = 11), and there are 11 letters in
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 to you or someone that you
Wisconsin has 5.4 million residents. On any given day, the probability is 1/5000 that a randomly selected Wisconsin resident decides to visit Disney World in Florida.a. Find the probability that they all will decide to go tomorrow, in which case Disney World has more than 5.4 million people in line
In the opening scene of Tom Stopper’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 flips, such that each outcome in
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, P(S) = 0.01 for the state of having breast cancer at a given time). The chance of breast cancer increases
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 truly has breast cancer. b. Show how
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 means that P(murdered by
In criminal trials (e.g., murder, robbery, driving while impaired, etc.) in the United States, it must be proven that a defendant is guilty beyond a reasonable doubt. This can be thought of as a very strong unwillingness to convict defendants who are actually innocent. In civil trials (e.g., breach
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 their DNA matching that found at the crime scene is
Example 8 about the Triple Blood Test for Down syndrome found the results shown in the table below.a. Estimate the probability that Down syndrome occurs (Down = Yes). b. Find the estimated (i) sensitivity and (ii) specificity. c. Find the estimated (i) P(Yes | POS) and (ii) P(No | NEG). d. Explain
The director of a local blood bank is in need of a donor with type AB blood. The distribution of blood types for Americans is estimated by the American Red Cross to beThe director decides that if more than 20 donors are required before the first donor with AB blood appears, she will need to issue a
In Example 16, we estimated the probability of winning at least $20 in the game was 0.40. Meanwhile, we concluded analytically that the actual probability was 0.20. a. Explain what caused the fairly large disparity between our estimated result and the actual result. b. The simulation in the example
In Example 16 and in the previous exercise, we used random numbers to estimate the probability of winning at least $20 in the game. Recall that to do so, for each repetition we simply recorded a Yes or No, pending whether or not we won at least $20 on the given repetition. At the end of the series
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. Does this mean that if we watch
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 that you had in the
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 dessert (Amelia’s
After the major accounting scandals with Enron, a large energy company, the question may be posed, €œWas there any way to examine Enron€™s accounting books to determine if they had been €˜doctored€™?€ One way uses Benford€™s law, which
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 married couple is
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 all 12 jurors are white
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 together? Using a tree
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 would you explain to someone
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. Given that a subject
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 once a month. Of
In the CASA teen survey described in the previous exercise, 33% of teens reported that parents are never present at parties they attend. Thirty-one percent of teens who say parents are never present during parties report that marijuana is available at the parties they attend, compared to only 1% of
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 product and tried the
A board game requires players to roll a pair of balanced dice for each player’s turn. Denote the outcomes by (die 1, die 2), such as (5, 3) for a 5 on die 1 and a 3 on die 2. a. List the sample space of the 36 possible outcomes for the two dice. b. Let A be the event that you roll doubles (that
Refer to the previous exercise. Define event D as rolling an odd sum with two dice. a. B denotes a sum of 7 on the two dice. Find P(B and D). When an event B is contained within an event D, as here, explain why P (B and D) = P (B). b. Find P(B or D). When an event B is contained within an event D,
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 projects are
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.
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 separate topics. On any given topic, there’s only a 0.02 probability that you agree. If you did agree on a topic, you would consider it to be
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. 10 As more and more evens occurred, the proportion of people betting on an odd outcome increased, as they figured
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 context. b. Jurors are asked to convict a
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 opening the smoker’s chest
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 review of published studies on the
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 will still have a positive hemoccult
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 proportion of
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 diagnostic
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 sensitivity was
A fast food chain is running a promotion to try to increase sales. Each customer who purchases a meal combo receives a game piece that contains one of the letters U, W, I, or N. If a player collects one of all four letters, that player is the lucky winner of a free milkshake. Suppose that of all
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 in order to win the
Using the Simulating the Probability of a Head With a Fair Coin applet on the text CD or other software, simulate the flipping of a balanced coin. a. Report the cumulative proportion of heads after (i) 10 flips, (ii) 100 flips, (iii) 1000 flips, and (iv) 10,000 flips. Explain how the results

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