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Statistics Concepts And Controversies 10th Edition David S. Moore, William I. Notz - Solutions
20.19 The Asian stochastic beetle again. In Exercise 20.17, you found the expected number of female offspring of the Asian stochastic beetle. Simulate the offspring of 100 beetles and find the mean number of offspring for these 100 beetles. Compare this mean with the expected value from Exercise
20.18 An expected rip-off? A “psychic” runs the following ad in a magazine:Expecting a baby? Renowned psychic will tell you the sex of the unborn child from any photograph of the mother. Cost, $20. Money-back guarantee.This may be a profitable con game. Suppose that the psychic simply replies
20.17 The Asian stochastic beetle. We met this insect in Exercise 19.21 (page 457). Females have this probability model for their number of female offspring:Offspring: 0 1 2 Probability: 0.2 0.3 0.5a. What is the expected number of female offspring?b. Use the law of large numbers to explain why the
20.16 Rolling two dice. Example 2 of Chapter 18 (page 428) gives a probability model for rolling two casino dice and recording the number of spots on each of the two up-faces. That example also shows how to find the probability that the total number of spots showing is five. Follow that method to
20.15 Keno. Keno is a popular game in casinos. Balls numbered 1 to 80 are tumbled in a machine as the bets are placed, then 20 of the balls are chosen at random. Players select numbers by marking a card. Here are two of the simpler Keno bets. Give the expected winnings for each.a. A $1 bet on
20.14 Estimating sales. Gain Communications sells aircraft communications units. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The sales manager estimates next year’s sales as
20.13 Making decisions. A six-sided die has two green and four red faces and is balanced so that each face is equally likely to come up. You must choose one of the following three sequences of colors:RGRRR RGRRRG GRRRRR Now start rolling the die. You will win $25 if the first rolls give the
20.12 Making decisions. The psychologist Amos Tversky did many studies of our perception of chance behavior. In its obituary of Tversky, the New York Times cited the following example.a. Tversky asked subjects to choose between two public health programs that affect 600 people. The first has
20.11 More roulette. An American roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. Gamblers bet on roulette by placing chips on a table that lays out the numbers and colors of the
20.10 More DC-3. There are other elaborate versions of DC-3 (Example 2). In the $1 Straight-Box (3-way) bet, if you choose a number with two digits the same, for example, 112, you win $330 if the randomly chosen winning number is 112, and you win $80 if the winning number has the digits 1, 1, and 2
20.9 DC-4. The “Straight” DC-4 lottery game is much like the “Straight” DC-3 game of Example 1. Winning numbers for both are reported on television and in local newspapers.You pay $1.00 and pick a four-digit number. The state chooses a four-digit number at random and pays you $5000 if your
20.8 The numbers racket. Pick 3 lotteries (Example 1) copy the numbers racket, an illegal gambling operation common in the poorer areas of large cities. States usually justify their lotteries by donating a portion of the proceeds to education. One version of a numbers racket operation works as
20.7 The law of large numbers. I simulate a random phenomenon that has numerical outcomes many, many times. If I average together all the outcomes I observe, this averagea. should be close to the probability of the random phenomenon.b. should be close to the expected value of the random
20.6 The law of large numbers. The law of large numbers says that the mean outcome in many repetitions of a random phenomenon having numerical outcomesa. gets close to the expected value as the number of repetitions increases.b. goes to zero as the number of repetitions increases because,
20.5 Expected value. You flip a coin for which the probability of heads is 0.5 and the probability of tails is 0.5. If the coin comes up heads, you win $1. Otherwise, you lose $1.The expected value of your winnings isa. $0.b. $1.c. −$1.d. $0.5.
20.4 Expected value. The expected value of a random phenomenon that has numerical outcomes isa. the outcome that occurs with the highest probability.b. the outcome that occurs more often than not in a large number of trials.c. the average of all possible outcomes.d. the average of all possible
20.3 Expected value. Which of the following is true of the expected value of a random phenomenon?a. It must be one of the possible outcomes.b. It cannot be one of the possible outcomes because it is an average.c. It can only be computed if the random phenomenon has numerical values.d. It is the
19.28 The multiplication rule. Here is another basic rule of probability: if several events are independent, the probability that all of the events happen is the product of their individual probabilities. We know, for example, that a child has probability 0.49 of being a girl and probability 0.51
19.27 The birthday problem. A famous example in probability theory shows that the probability that at least two people in a room have the same birthday is already greater than 1-in-2 when 23 people are in the room. The probability model is 1. The birth date of a randomly chosen person is equally
19.26 More on overbooking. Let’s continue the simulation of Exercise 19.24. You offer special vouchers to people who willingly give up their seats when the plane is overbooked.The probability that a passenger will volunteer to accept a special voucher when a plane is overbooked is 0.2. You want
19.25 A multiple-choice exam. Matt has lots of experience taking multiple-choice exams without doing much studying. He is about to take a quiz that has 10 multiple-choice questions, each with four possible answers. Here is Matt’s personal probability model. He thinks that in 75% of questions, he
19.24 Overbooking. Your company operates small commuter planes. Each plane carries eight passengers. Some passengers who reserve seats don’t show up—in fact, the probability is 0.1 that a randomly chosen passenger will fail to appear. Passengers’ behaviors are independent. If you allow nine
19.23 Playing craps. The game of craps is played with two dice. The player rolls both dice and wins immediately if the outcome (the sum of the faces) is 7 or 11. If the outcome is 2, 3, or 12, the player loses immediately. If he rolls any other outcome, he continues to throw the dice until he
19.22 Two warning systems. An airliner has two independent automatic systems that sound a warning if there is terrain ahead (that means the airplane is about to fly into a mountain).Neither system is perfect. System A signals in time with probability 0.9. System B does so with probability 0.8. The
19.21 The Asian stochastic beetle. We can use simulation to examine the fate of populations of living creatures. Consider the Asian stochastic beetle. Females of this insect have the following pattern of reproduction:1. 20% of females die without female offspring, 30% have one female offspring, and
19.20 Gambling in ancient Rome. Tossing four astragali was the most popular game of chance in Roman times. Many throws of a present-day sheep’s astragalus show that the approximate probability distribution for the four sides of the bone that can land uppermost are Outcome Probability Narrow flat
19.19 A better model for repeating an exam. A more realistic probability model for Elaine’s attempts to pass an exam in the previous exercise is as follows. On the first try, she has probability 0.4 of passing. If she fails on the first try, her probability on the second try increases to 0.5
19.18 Repeating an exam. Elaine is enrolled in a self-paced course that allows three attempts to pass an examination on the material. She skims the online reading and then takes the exam. Assume that after only skimming the online material, Elaine has probability 0.4 of passing on any one attempt.
19.17 Sue’s three-point shooting. Sue Bird of the Women’s National Basketball Association team Seattle Storm makes 39% of her three-point shots. In an important game, she shoots four three-point shots late in the game and misses all of them. The fans think she was nervous, but the misses may
19.16 LeBron’s’s three-point shooting. The basketball player LeBron James makes about 34% of his three-point shots over an entire season. Take his probability of a success to be 0.34 on each shot. Using line 122 of Table A, simulate 25 repetitions of his performance in a game in which he shoots
19.15 More on first-year college students. In Exercise 19.13, you explained how to simulate the response of a randomly chosen first-year college student to the question of how many hours during a typical week they spent studying or doing homework during their last year in high school. The Tutoring
19.14 More on an easy A. In Exercise 19.12, you explained how to simulate the grade of a randomly chosen student who took the accelerated statistics course in the last 10 years.Suppose you select five students at random who took the course in the last 10 years. Use simulation to estimate the
19.13 First-year college students. Select a first-year college student at random and ask how many hours during a typical week did they spend studying or doing homework during their last year in high school? Probabilities for the outcomes are Time: Less than one hour 1 to 5 hours 6 to 10 hours More
19.12 An easy A? Choose a student at random from all who took the large accelerated introductory statistics courses at Hudson River College in the last 10 years. The probabilities for the student’s grade are Grade: A B C D or F Probability: 0.4 0.3 0.2 ?a. What must be the probability of getting
19.11 Simulating an opinion poll. A 2018 poll by the Pew Research Survey Center interviewed a random sample of 2002 adult Americans. Those in the sample were asked which president has done the best job in their lifetime. The poll showed that about 30% of adult Americans regarded Barack Obama as
19.10 Basic simulation. Use Table A to simulate the responses of 10 independently chosen adults in each of the four situations of Exercise 19.8.a. For situation (a), use line 110.b. For situation (b), use line 111.c. For situation (c), use line 112.d. For situation (d), use line 113.
19.9 On time flights. Suppose that 80% of American Airlines flights are on time (this is approximately the percentage of American Airline flights on time in November 2018). You check 10 American Airline flights chosen at random. What is the probability that all 10 are on time?a. Give a probability
19.8 Approval ratings of Democrats in Congress. An opinion poll selects adult Americans at random and asks them, “Do you approve or disapprove of the way Democrats in Congress are handling their job?” Explain carefully how you would assign digits from Table A to simulate the response of one
19.7 Elaborate simulations. The key to successful simulation isa. keeping the tree diagram as simple as possible.b. thinking carefully about the probability model for the simulation.c. using as few trials as possible so that the chance of an incorrect trial is kept small.d. using all the digits in
19.6 Elaborate simulations. A tree diagrama. was originally used by biologists for simulations involving trees.b. is used to determine if two random phenomena are independent.c. is used when two random phenomena are independent.d. specifies a probability model in graphical form.
19.5 Independence. Two random phenomena are independent ifa. knowing that one of the outcomes has occurred means the other cannot occur.b. knowing the outcomes of one does not change the probabilities for outcomes of the other.c. both have the same probability of occurring.d. both have different
19.4 A simulation. To simulate the toss of a fair coin (the probability of heads and tails are both 0.5) using a table of random digits,a. assign the digits 0, 1, 2, 3, and 4 to represent heads and the digits 5, 6, 7, 8, and 9 to represent tails.b. assign the digits 0, 2, 4, 6, and 8 to represent
19.3 Simulations. To simulate random outcomes, we need to knowa. the probabilities of the outcomes.b. whether the probabilities are personal probabilities.c. both the mean and standard deviation so we can use the appropriate Normal curve.d. that the random outcome has a probability of 0.1 so we can
18.28 The addition rule (optional). Probability rule D states: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. This is sometimes called the addition rule for disjoint events. A more general form of the addition
18.27 Odds and personal probability. One way to determine your personal probability about an event is to ask what you would consider a fair bet that the event will occur. Suppose in August 2018 you believed it fair that a bet of $2 should win $10 if the Philadelphia Eagles win Super Bowl 53.a. What
18.26 Generating a sampling distribution. Let us illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam:Student: 0 1 2 3 4 5 6 7 8 9 Score: 82 62 80 58 72 73 65 66 74 62 The parameter of
18.25 Applying to college. You ask an SRS of 1500 college students whether they applied for admission to any other college. Suppose that, in fact, 35% of all college students applied to colleges besides the one they are attending. (That’s close to the truth.) The sampling distribution of the
18.24 Do you jog? An opinion poll asks an SRS of 1500 adults, “Do you jog?” Suppose (as is approximately correct) that the population proportion who jog is p=0.20 In a large number of samples, the proportion p^ who answer Yes will be approximately Normally distributed with mean 0.20 and
18.23 Airplane safety (optional). In the setting of Exercise 18.21, what is the probability of getting a sample in which more than 45% think that airline travel is safer than driving? (Use Table B.)
18.22 Immigration (optional). In the setting of Exercise 18.20, what is the probability of getting a sample in which more than 30% of those sampled think that the level of immigration to this country should be decreased? (Use Table B.)
18.21 Airplane safety. Suppose that 44% of all adults think that airline travel is safer than driving. An opinion poll plans to ask an SRS of 1021 adults about airplane safety. The proportion of the sample who think that airline travel is safer than driving will vary if we take many samples from
18.20 Immigration. Suppose that 29% of all adult Americans think that the level of immigration to the United States should be decreased. An opinion poll interviews 1520 randomly chosen Americans and records the sample proportion who feel that the level of immigration to this country should be
18.19 Legitimate probabilities? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, whether it satisfies the rules of probability. If not, give specific reasons for your answer.a. When a coin is spun, P( H
18.18 Colors of M&Ms. If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made.a. Here are the probabilities of each color for a randomly chosen
18.17 Roulette. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and, at the same time, rolls a small ball along the wheel in the opposite direction. The wheel is carefully
18.16 More tetrahedral dice. Tetrahedral dice are described in Exercise 18.14. Give a probability model for rolling two such dice. That is, write down all possible outcomes and give a probability to each. (Example 2 and Figure 18.1 may help you.) What is the probability that the sum of the
18.15 Birth order. A couple plan to have three children. There are eight possible arrangements of girls and boys. For example, GGB means the first two children are girls and the third child is a boy. All eight arrangements are (approximately) equally likely.a. Write down all eight arrangements of
18.14 Tetrahedral dice. Psychologists sometimes use tetrahedral dice to study our intuition about chance behavior. A tetrahedron (Figure 18.6) is a pyramid with four faces, each a triangle with all sides equal in length. Label the four faces of a tetrahedral die with one, two, three, and four
18.13 Political views of college students. Select a first-year college student at random and ask how they would characterize their political views. Here are the probabilities, based on proportions from a large sample survey in 2016 of first-year students:Political view: Far left Liberal Middle of
18.12 Rolling a die. Figure 18.5 displays several assignments of probabilities to the six faces of a die. We can learn which assignment is actually correct for a particular die only by rolling the die many times. However, some of the assignments are not legitimate assignments of probability. That
18.11 Our favorite president. A 2018 poll by the Pew Research Survey Center interviewed a random sample of 2002 adult Americans. Those in the sample were asked which president has done the best job in their lifetime. Here are the results:Outcome Probability Barack Obama 0.31 Ronald Reagan 0.21 Bill
18.10 Land in Canada. Choose an acre of land in Canada at random. The probability is 0.38 that it is forest and 0.07 that it is agricultural.a. What is the probability that the acre chosen is not forested?b. What is the probability that it is either forest or agricultural?c. What is the probability
18.9 Causes of death. Government data assign a single cause for each death that occurs in the United States. Data from 2016 show that the probability is 0.23 that a randomly chosen death was due to heart disease and 0.22 that it was due to cancer. What is the probability that a death was due either
18.8 Moving up. Sociologists studying social mobility in the United States find that the probability that someone who began their career in the bottom 10% of earnings remains in the bottom 10% 15 years later is 0.59. What is the probability that such a person moves to one of the higher income
18.7 Sampling distributions. The sampling distribution of a statistic isa. the method of sampling used to obtain the data from which the statistic is computed.b. the possible methods of computing a statistic from the data.c. the pattern of the data from which the statistic is computed.d. the
18.6 Probability rules. To find the probability of any event,a. add up the probabilities of the outcomes that make up the event.b. use the probability of the outcome that best approximates the event.c. assign it a random, but plausible, value between 0 and 1.d. average together the personal
18.5 Density curves. Which of the following is true of density curves?a. Areas under a density curve determine probabilities of outcomes.b. The total area under a density curve is 1.c. The Normal curve is a density curve.d. All of the above are true.
18.4 Probability models. Which of the following is true of any legitimate probability model?a. The probabilities of the individual outcomes must be numbers between 0 and 1, and they must sum to no more than 1.b. The probabilities of the individual outcomes must be numbers between 0 and 1, and they
18.3 Probability models. A probability model describes a random phenomenon by telling us which of the following?a. Whether we are using data-based or personal probabilities.b. What outcomes are possible and how to assign probabilities to these outcomes.c. Whether the probabilities of all outcomes
17.32 What probability doesn’t say. The probability of a head in tossing a coin is 1-in-2.This means that as we make more tosses, the proportion of heads will eventually get close to 0.5. It does not mean that the count of heads will get close to one-half the number of tosses.To see why, imagine
17.31 Reacting to risks. National newspapers such as USA Today and the New York Times carry many more stories about deaths from airplane crashes than about deaths from motor vehicle crashes. Motor vehicle accidents killed about 32,700 people in the United States in 2013. Crashes of all scheduled
17.30 Reacting to risks. The probability of dying if you play high school football is about 10 per million each year you play. The risk of getting cancer from asbestos if you attend a school in which asbestos is present for 10 years is about 5 per million. If we ban asbestos from schools, should we
17.29 An unenlightened gambler.a. A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes five consecutive reds occur and bets heavily on black on the next spin. Asked why, he explains that black is “due by the law of averages.” Explain to
17.28 Snow coming. A meteorologist, predicting below-average snowfall this winter, says,“First, in looking at the past few winters, there has been above-average snowfall. Even though we are not supposed to use the law of averages, we are due.” Do you think that “due by the law of averages”
17.27 The “law of averages.” The baseball player Jose Altuve gets a hit about 31.6% of the time over an entire season. After he has failed to hit safely in nine straight at-bats, the TV commentator says, “Jose is due for a hit by the law of averages.” Is that right? Why?
17.26 In the long run. Probability works not by compensating for imbalances but by overwhelming them. Suppose that the first 10 tosses of a coin give 10 tails and that tosses after that are exactly half heads and half tails. (Exact balance is unlikely, but the example illustrates how the first 10
17.25 Curry’s free throws. The basketball player Stephen Curry is the all-time career freethrow shooter among active players. He makes 90.4% of his free throws. In today’s game, Curry misses his first two free throws. The TV commentator says, “Curry’s technique looks out of rhythm today.”
17.24 An eerie coincidence? A September 13, 2011, New York Post article reported that the first three winners at Belmont Park were horses wearing the numbers 9, 1, 1 on the tenth-year anniversary of the 9/11 attacks on America. Should this fact surprise you? Explain your answer.
17.23 Surprising? During the Michigan versus Ohio State football game in 2018, news media reported that Jim Harbaugh and Urban Meyer, the head coaches of Michigan and Ohio State, respectively, were born in the same hospital in Toledo, Ohio. That a pair of coaches from two arch rivals were also born
17.22 Playing Pick 4. The Pick 4 games in many state lotteries announce a four-digit winning number each day. The winning number is essentially a four-digit group from a table of random digits. You win if your choice matches the winning digits, in exact order. The winnings are divided among all
17.21 Personal random numbers? Ask several of your friends (at least 10 people) to choose a four-digit number “at random.” How many of the numbers chosen start with 1 or 2? How many start with 8 or 9? (There is strong evidence that people in general tend to choose numbers starting with low
17.20 Personal probability? When there are few data, we often fall back on personal probability. There had been just 24 space shuttle launches, all successful, before the Challenger disaster in January 1986. The shuttle program management thought the chances of such a failure were only 1 in
17.19 Personal probability versus data. Give an example in which you would rely on a probability found as a long-term proportion from data on many trials. Give an example in which you would rely on your own personal probability.
17.18 Marital status. Based on 2018 data, the probability that a randomly chosen woman over 64 years of age is divorced is about 0.14. This probability is a long-run proportion based on all the millions of women over 64. Let’s suppose that the proportion stays at 0.14 for the next 45 years.
17.17 Will you have an accident? The probability that a randomly chosen driver will be involved in an accident in the next year is about 0.3. This is based on the proportion of millions of drivers who have accidents. “Accident” includes things like crumpling a fender in your own driveway, not
17.16 Winning a baseball game. Over the period from 1965 to 2018, the champions of baseball’s two major leagues won 63% of their home games during the regular season. At the end of each season, the two league champions meet in the baseball World Series. Would you use the results from the regular
17.15 From words to probabilities. Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given.(The probability is usually a more exact measure of likelihood than is the verbal statement.)0 0.01 0.4 0.6 0.99 1a.
17.14 Four-of-a-kind. You read in a book on poker that the probability of being dealt fourof-a-kind (a hand containing four cards of the same value, such as four kings) in a five-card poker hand is about 0.00024. Explain in simple language what this means.
17.13 Rolling dice. Roll a pair of dice 100 times. How many times did you roll a 5? What is the approximate probability of rolling a 5?
17.12 Tossing a thumbtack. Toss a thumbtack on a hard surface 50 times. How many times did it land with the point up? What is the approximate probability of landing point up?
17.11 How many tosses to get a head? When we toss a penny, experience shows that the probability (long-term proportion) of a head is close to 1-in-2. Suppose now that we toss the penny repeatedly until we get a head. What is the probability that the first head comes up in an even number of tosses
17.10 Random digits. The table of random digits (Table A) was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the 400 digits in lines 120 to 129 in the table are 0s? This proportion is an estimate, based on 400 repetitions, of the true
17.9 Nickels falling over. You may feel that it is obvious that the probability of a head in tossing a coin is about 1-in-2 because the coin has two faces. Such opinions are not always correct. The previous exercise asked you to spin a nickel rather than toss it—that changes the probability of a
17.8 Nickels spinning. Hold a nickel upright on its edge under your forefinger on a hard surface, then snap it with your other forefinger so that it spins for some time before falling.Based on 50 spins, estimate the probability of heads.
17.7 Personal probabilities. Which of the following is true of a personal probability about the outcome of a phenomenon?a. It expresses an individual’s judgment of how likely an outcome is.b. It can be any number because personal probabilities need not be restricted to values between 0 and 1.c.
17.6 Probability. I toss a coin 1000 times and observe the outcome “heads” 481 times.Which of the following can be concluded from this result?a. This is suspicious because we should observe exactly 500 heads if the coin is tossed 1000 times.b. The probability of heads is approximately 481.c.
17.5 Probability. Which of the following is true of probability?a. It is a number between 0 and 1.b. A probability of 0 means the outcome never occurs.c. A probability of 1 means the outcome always occurs.d. All of the above are true.
17.4 Probability. The probability of a specific outcome of a random phenomenon isa. the number of times it occurs in very many repetitions of the phenomenon.b. the number repetitions of the phenomenon it takes for the outcome to first occur.c. the proportion of times it occurs in very many
17.3 Randomness. Random phenomena have which of the following characteristics?a. They must be natural events. Man-made events cannot be random.b. They exhibit a clear pattern in very many repetitions, although any one trial of the phenomenon is unpredictable.c. Future outcomes must compensate for
3. Write a paragraph discussing whether the “surprising” coincidence described in the Case Study that began this chapter is as surprising as it might first appear.
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