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nonparametric statistical inference
Statistical Reasoning For Everyday Life 4th Edition Jeffrey Bennett - Solutions
Behind in Coin Tossing: Can you Catch up? Suppose that you toss a fair coin 100 times, getting 38 heads and 62 tails, which is 24 more tails than heads.a. Explain why, on your next toss, the difference in the numbers of heads and tails is as likely to grow to 25 as it is to shrink to 23.b. Extend
gambler’s Fallacy and Dice. Suppose you roll a die with a friend, with the following rules: For every even number you roll, you win $1 from your friend; for every odd number you roll, you pay $1 to your friend.a. What are the chances of rolling an even number on one roll of a fair die? An odd
Expected Value for a magazine Sweepstakes. Reader’s Digest ran a sweepstakes in which prizes were listed along with the chances of winning: $1,000,000 (1 chance in 90,000,000), $100,000 (1 chance in 110,000,000), $25,000(1 chance in 110,000,000), $5,000 (1 chance in 36,667,000), and $2,500 (1
Psychology of Expected Values. In 1953, a French economist named Maurice Allais conducted a survey of how people assess risk. Here are two scenarios that he used, each of which required people to choose between two options.Decision 1:Option A: 100% chance of gaining $1,000,000 Option B: 10% chance
New Jersey Pick 4.In New Jersey’s Pick 4 lottery game, you pay 50¢ to select a sequence of four digits, such as 2273, from the 10,000 different possible four-digit sequences. If you select the same sequence of four digits that are drawn, you win and collect $2788. What is your expected value? In
Expected Value in Casino Dice. When you give a casino$5 for a bet on the “pass line” in a casino game of dice,there is a 251/495 probability that you will lose $5 and there is a 244/495 probability that you will make a net gain of $5. (If you win, the casino gives you $5 and you get to keep
Expected Value in Roulette. When you give the Venetian casino in Las Vegas $5 for a bet on the number 7 in roulette, you have a 37/38 probability of losing $5 and you have a 1/38 probability of making a net gain of $175. (The prize is $180, but your $5 bet is not returned, so the net gain is $175.)
Big game. The Multi-State Big Game lottery advertises the following prizes and probabilities of winning for a single $1 ticket. The jackpot is variable, but assume it has an average value of $3 million. Note that the same prize can be given to two outcomes with different probabilities.What is the
Powerball Lottery. The multi-state Powerball lottery advertises the following prizes and probabilities of winning for a single $1 ticket. Assume the jackpot has a value of$30 million one week. Note that there is more than one way to win some of the monetary prizes (for example, two ways to win
Expected Waiting Time. Suppose that you arrive at a bus stop randomly, so all arrival times are equally likely. The bus arrives regularly every 30 minutes without delay (say, on the hour and on the half hour). What is the expected value of your waiting time? Explain how you got your answer.
Insurance Claims. An actuary at an insurance company estimates from existing data that on a $1,000 policy, an average of 1 in 100 policyholders will file a $20,000 claim, an average of 1 in 200 policyholders will file a $50,000 claim, and an average of 1 in 500 policyholders will file a$100,000
Extra Points in Football. Football teams have the option of trying to score either 1 or 2 extra points after a touchdown. They get 1 point by kicking the ball through the goal posts or 2 points by running or passing the ball across the goal line. For a recent year in the NFL, 1-point kicks were
Kentucky’s Pick 4 Lottery. If you bet $1 in Kentucky’s Pick 4 lottery game, you either lose $1 or gain $4,999. (The winning prize is$5,000, but your $1 bet is not returned, so the net gain is $4,999.) The game is played by selecting a four-digit number between 0000 and 9999. What is the
Should you Play? Suppose someone offers you this opportunity:You can place a bet of $10 and roll a single die once. You win twice the outcome of the die. For example, if you roll a 6, you win $12 for a net profit of $2. What is the expected value of this game? Should you play?
Speedy Driver. A person who has a habit of driving fast has never had an accident or traffic citation. What does it mean to say that “the law of averages will catch up with him”? Is it true? Explain.
gender Selection. In planning an experiment designed to test the effectiveness of a gender selection method, assume that boys and girls are equally likely to be born. Among 500 births, should we expect to get exactly 250 boys and 250 girls? As the number of births increases, what does the law of
Lottery. Jennifer purchases a state lottery ticket and she avoids the combination of 1, 2, 3, 4, 5, and 6.She reasons that this combination has the same chance as any other combination, but the jackpot is divided among the winners, and many people are likely to select 1, 2, 3, 4, 5, and 6, so the
Lottery. When considering the chances of winning a lottery, Kim reasons that the number combination of 1, 2, 3, 4, 5, and 6 is less likely to occur because of the obvious pattern.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain
Lottery. Kelly studies the Wisconsin Pick 3 lottery and finds that for a $1 bet, the expected return is 50¢, so she reasons that this is bad bet and she does not play.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain clearly; not
Gambling Strategy. Steve learns that for a $5 bet on a number in roulette, the expected return is $4.74, but for a$5 bet on the pass line in the game of craps, the expected return is $4.93, so it is better to play this particular craps bet than the roulette bet.decide whether the statement makes
Gambler’s Fallacy. In your own words, describe the gambler’s fallacy.
Gambling Strategy. A professional gambler playing blackjack in the Venetian casino has lost each of his first 10 bets.He begins to place larger bets, reasoning that his current proportion of wins (which is 0) will increase to get closer to the average number of wins. Is his betting strategy
Expected Value. A geneticist computes the expected number of girls in 5 births and obtains the result of 2.5 girls. He rounds the result to 3 girls, reasoning that it is impossible to get 2.5 girls in 5 births. Is that reasoning correct? Why or why not?
Law of Large Numbers. In your own words, describe the law of large numbers.
Probability Distributions. Find a news article or research report that cites or makes use of a probability distribution. Provide a one-paragraph discussion.
Subjective Probabilities. Find a news article or research report that refers to a subjective probability.Provide a one-paragraph discussion.
Relative Frequency Probabilities. Find a news article or research report that makes use of a relative frequency (or empirical) probability. Provide a oneparagraph discussion.
Theoretical Probabilities. Find a news article or research report that cites a theoretical probability.Provide a one-paragraph discussion.
Randomizing a Survey. Suppose you want to conduct a survey involving a sensitive question that not all participants may choose to answer honestly (for example, a question involving cheating on taxes or drug use). Here is a way to conduct the survey and protect the identity of respondents. We will
Three-Coin Experiment. Toss three coins at once 50 times and record the outcomes in terms of the number of heads. Based on your observations, give the relative frequency probabilities of the outcomes. Do they agree with the theoretical probabilities? Explain and discuss your results.
Thumb Tack Probabilities. Find a standard thumb tack and practice tossing it onto a flat surface. Notice that there are two different outcomes: The tack can land point down or point up.a. Toss the tack 50 times and record the outcomes.b. Give the relative frequency probabilities of the two outcomes
age and gender. The proportions of men and women in the population change with age. Using current data from a Web site, construct a table showing the probability of meeting a male or a female in each of these age categories: 0–5, 6–10, 11–20, 21–30, 31–40, 41–50, 51–60, 61–70,
Blood groups. The four major blood groups are designated A, B, AB, and O. Within each group there are two Rh types: positive and negative. Using library resources or the Internet, find data on the relative frequency of blood groups, including the Rh types. Construct a table showing the probability
Colorado Lottery Distribution. The histogram in Figure 7 shows the distribution of 5,964 Colorado lottery numbers(possible values range from 1 to 42).a. Assuming the lottery drawings are random, what would you expect the probability of any number to be?b. Based on the histogram, what is the
Four-Coin Probability Distribution.a. Construct a table similar to Table 2, showing all possible outcomes of tossing four coins at once.b. Construct a table similar to Table 3, showing the probability distribution for the events 4 heads, 3 heads, 2 heads, 1 head, and 0 heads when you toss four
age at First marriage. The following table gives percentages of women and men married for the first time in several age categories (U.S. Census Bureau)a. What is the probability that a randomly encountered married woman was married, for the first time, between the ages of 35 and 44?b. What is the
Senior Citizen Probabilities. In the year 2000, there were 34.7 million people over 65 years of age out of a U.S. population of 281 million. In the year 2050, it is estimated that there will be 78.9 million people over 65 years of age out of a U.S. population of 394 million. Would your chances of
Surgery. In a clinical trial of 73 carpal tunnel syndrome patients treated with surgery, 67 had successful treatments(based on data from “Splinting vs. Surgery in the Treatment of Carpal Tunnel Syndrome” by Gerritsen et al., Journal of the American Medical Association, Vol. 288, No. 10). What
Basketball. Halfway through the season, a basketball player has hit 86% of her free throws. What is the probability that her next free throw will be successful?Use the relative frequency method to estimate the probabilities.
Flood. What is the probability of a 100-year flood this year?Use the relative frequency method to estimate the probabilities.
Weather Forecast. After recording the forecasts of your local weatherman for 30 days, you conclude that he gave a correct forecast 18 times. What is the probability that his next forecast will be correct?Use the relative frequency method to estimate the probabilities.
Four-Child Family. Suppose you randomly select a family with four children. Assume that births of boys and girls are equally likely.a. How many birth orders are possible? List all of them.b. What is the probability that the family has four boys?Four girls?c. What is the probability that the family
Three-Child Family. Suppose you randomly select a family with three children. Assume that births of boys and girls are equally likely. What is the probability that the family has each of the following?a. Three girlsb. Two boys and a girlc. A girl, a boy, and a boy, in that orderd. At least one
Test Questions. The New England College of Medicine uses an admissions test with multiple-choice questions, each with five possible answers, only one of which is correct. If you guess randomly on every question, what score might you expect to get? (Express the answer as a percentage.)use the
m&ms. A bag contains 10 red M&Ms, 15 blue M&Ms, and 20 yellow M&Ms. What is the probability of drawing a red M&M? A blue M&M? A yellow M&M? Something besides a yellow M&M?use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to
Defects. What is the probability of not getting a defective fuse when one fuse is randomly selected from an assembly line and 1% of the fuses are defective?find the probability of the given event. State any assumptions that you use.
Baseball. What is the probability that a 0.280 hitter in baseball will not get a hit on his next at-bat?find the probability of the given event. State any assumptions that you use.
Testing. What is the probability of guessing incorrectly when making a random guess on a multiple-choice test question with possible answers ofa, b,c, d, ande, one of which is correct?
Basketball. What is the probability that a 55% free-throw shooter will miss her next free throw?find the probability of the given event. State any assumptions that you use.
Birthday. What is the probability of finding that the next President of the United States was not born on Saturday?find the probability of the given event. State any assumptions that you use.
Week Days. What is the probability of randomly selecting a day of the week and not getting Monday?find the probability of the given event. State any assumptions that you use.
Die. What is the probability of rolling a fair die and not getting an outcome that is greater than 6?find the probability of the given event. State any assumptions that you use.
Die. What is the probability of rolling a fair die and not getting an outcome less than 7?find the probability of the given event. State any assumptions that you use.
Dice. Rolling a pair of dice and getting an outcome (sum)of 12.use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Births. Finding that the next baby born to a couple is a girl, given that the couple already has two children and they are both boys.use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Births. Finding that the next baby born in Alaska is a girl.use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Birthday. Finding that the next person you meet has the same birthday as yours (Ignore leap years.).use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Birthday. Finding that the next President of the United States was born on Saturday.use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Roulette. Getting an outcome of a red slot when a roulette wheel is spun (A roulette wheel has slots of 0, 00, 1, 2, 3, . . . , 36, and 18 of those slots are red.).use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Test Question. Making a correct random guess for an answer to a particular multiple-choice question with possible answers ofa, b,c, d,e, one of which is correct.use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Die. Rolling a die and getting an outcome that is greater than 2.use the theoretical method to determine the probability of the given outcome or event. State any assumptions that you need to make.
Lightning. Jill estimates that the subjective probability of her being struck by lightning sometime next year is 1/1,000,000.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false).Explain clearly; not all of these have definitive answers, so your
Lightning. Jack estimates that the subjective probability of his being struck by lightning sometime next year is 1/2.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false).Explain clearly; not all of these have definitive answers, so your
Car Crash. An insurance company states that the probability that a particular car will be involved in a car crash this year is 0.6 and the probability that the car will not be involved in a car crash this year is 0.3.decide whether the statement makes sense (or is clearly true) or does not make
Complementary Events. If there is a 0.9 probability that it will rain sometime today, then there is a probability of 0.1 that it will not rain sometime today.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false).Explain clearly; not all of these
Impossible Event. Because it is impossible for Thanksgiving to fall on Tuesday, the probability of Thanksgiving falling on Tuesday is 0.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false).Explain clearly; not all of these have definitive
Certain Event. When randomly selecting a day of the week, it is certain that you will select a day containing the letter y, so P(y) = 1.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false).Explain clearly; not all of these have definitive
Subjective Probability. Use subjective judgment to estimate the probability that the next time you ride an elevator, it gets stuck between floors.
Interpreting Probability. What do we mean when we say that “the probability of getting 20 babies of the same gender when 20 random babies are born is 1/524,288? Is such an event unusual? Why or why not?
Probability of Life. A student reasons that there is a probability of 1/2 that there is life on Neptune, because there are two possible events: There is life or there is not life. Is this reasoning correct? Why or why not?
Notation. If A denotes the event that you answer a particular true/false test question correctly, what do each of the following represent: P(A), P(not A), P(A), and what are their values?
Personal Statistical Significance. Describe an incident in your own life that did not meet your expectation, defied the odds, or seemed unlikely to have occurred by chance. Would you call this incident statistically significant? To what did you attribute the event?
Significant Experiment? Find a recent news story about a statistical study that used an experiment to determine whether some new treatment was effective.Based on the available information, briefly discuss what you can conclude about the statistical significance of the results. Given this
Statistical Significance. Find a recent newspaper article on a statistical study in which the idea of statistical significance is used. Write a one-page summary of the study and the result that is considered to be statistically significant. Also include a brief discussion of whether you believe the
Lengths of Rivers. Using an almanac or the Internet, find the lengths of the principal rivers of the world. Construct a list of the leading digits only. Does any particular digit occur more often than the others? Does that digit occur significantly more often? Explain.
Significance in Vital Statistics. Visit a Web site that has vital statistics (for example, the U.S. Census Bureau or the National Center for Health Statistics). Choose a question such as one of the following:• Are there significant differences in numbers of births among months?• Are there
Weight by age. A National Health Survey determined that the mean weight of a sample of 804 men ages 25 to 34 years was 176 pounds, while the mean weight of a sample of 1,657 men ages 65 to 74 years was 164 pounds.The difference is significant at the 0.01 level. Interpret this result.
SaT Preparation. A study of 75 students who took an SAT preparation course (American Education Research Journal, Vol. 19, No. 3) concluded that the mean improvement on the SAT was 0.6 point. If we assume that the preparation course has no effect, the probability of getting a mean improvement of 0.6
Seat Belts and Children. In a study of children injured in automobile crashes (American Journal of Public Health, Vol. 82, No. 3), those wearing seat belts had a mean stay of 0.83 day in an intensive care unit. Those not wearing seat belts had a mean stay of 1.39 days. The difference in means
human Body Temperature. In a study by researchers at the University of Maryland, the body temperatures of 106 individuals were measured; the mean for the sample was 98.20°F. It is commonly believed that the mean body temperature is 98.60°F. The difference between the sample mean and the accepted
Bednets and malaria. In a randomized controlled trial in Kenya, insecticide-treated bednets were tested as a way to reduce malaria. Among 343 infants who used the bednets, 15 developed malaria. Among 294 infants not using bednets, 27 developed malaria (based on data from“Sustainability of
gender Selection. The Genetics and IVF Institute conducted a clinical trial of its method for gender selection.The latest actual results showed that among 945 babies born to couples using the XSORT method of gender selection, 879 were girls.
Carpal Tunnel Syndrome Treatments. An experiment was conducted to determine whether there is a difference between the success rates from treating carpal tunnel syndrome with surgery and with splinting. The success rate for 73 patients treated with surgery was 92%, and the success rate for 83
Fuel Tests. Thirty identical cars are selected for a fuel test. Half of the cars are filled with regular gasoline, and the other half are filled with a new experimental fuel. The cars in the first group average 29.3 miles per gallon, while the cars in the second group average 35.5 miles per gallon.
Clinical Trial. In a clinical trial of a new drug intended to treat allergies, 5 of the 80 subjects in the treatment group experienced headaches, and 8 of the 160 subjects in the control group experienced headaches.state whether the difference between what occurred and what you would have expected
Jury Composition. For a trial on a charge of failure to pay child support, the jury consists of exactly 6 men and 6 women.state whether the difference between what occurred and what you would have expected by chance is statistically significant.Discuss any implications of the statistical
Subway Riders. A commuter enters a New York City subway car near Times Square and finds that it is occupied by 50 men, all of whom are bald.state whether the difference between what occurred and what you would have expected by chance is statistically significant.Discuss any implications of the
Survey. In conducting a survey of adults in the United States, a pollster claims that he randomly selected 20 subjects and all of them were women.state whether the difference between what occurred and what you would have expected by chance is statistically significant.Discuss any implications of
Lottery. A common lottery game is to select four digits, each between 0 and 9 (inclusive). Winning requires that you get the same four digits that are drawn, and they must be in the same order. You select four digits for one ticket, and none of your selections matches the numbers drawn.state
Rolls of a Die. In 6 rolls of a six-sided die, the outcome of 6 never occurs.state whether the difference between what occurred and what you would have expected by chance is statistically significant.Discuss any implications of the statistical significance.
Voter Survey. In a pre-election survey, 100 likely voters are randomly selected from adults in the United States, and it is found that 6% of them are Democrats.state whether the difference between what occurred and what you would have expected by chance is statistically significant.Discuss any
Statistics Class Survey. In a survey, 50 students are randomly selected among all college students currently taking a statistics course, and it is found that they are all females.state whether the difference between what occurred and what you would have expected by chance is statistically
Clinical Trial. In a clinical trial of a treatment for reducing back pain, the difference between the treatment group and the control group (with no treatment) was found to be statistically significant. This means that the treatment will definitely ease back pain for everyone.decide whether the
gender Selection. In a test of a technique of gender selection, the 100 babies born consist of at least 80 girls. Because there is about 1 chance in a billion of getting at least 80 girls among 100 babies, the results are statistically significant.decide whether the statement makes sense (or is
Statistical Significance. In an experiment testing a method of gender selection, 1,000 couples give birth to 550 girls and 450 boys. Because the probability of getting such extreme results by chance is only 0.0009, the results are statistically significant.decide whether the statement makes sense
unemployment. The rate of unemployment has statistical significance because it has such a strong effect on the economy.decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain clearly; not all of these have definitive answers, so your
Statistical Significance. If a particular result is statistically significant at the 0.01 level, must it also be statistically significant at the 0.05 level? Why or why not?
Statistical Significance. If a particular result is statistically significant at the 0.05 level, must it also be statistically significant at the 0.01 level? Why or why not?
Statistical Significance. Does the term statistical significance refer to results that are significant in the sense that they have great importance? Explain.
Statistical Significance. In an experiment testing a method of gender selection intended to increase the likelihood that a baby is a girl, 1,000 couples give birth to 501 girls and 499 boys. A company representative argues that this is evidence that the method is effective, because the probability
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