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essentials of statistics
The Basic Practice Of Statistics 5th Edition David S Moore - Solutions
Estimating π from random numbers. Kenyon College student Eric Newman used basic geometry to evaluate software random number generators as part of a summer research project. He generated 2000 independent random points (X, Y )in the unit square. (That is, X and Y are independent random numbers
A mixed group: probabilities.We would like to find the probability that exactly 2 of the 20 exposed children in the previous exercise develop whooping cough.(a) One way to get 2 infections is to get 1 among the 17 vaccinated children and 1 among the 3 unvaccinated children. Find the probability of
A mixed group: means. A group of 20 children at a nursery school are exposed to whooping cough by playing with an infected child. Of these children 17 have been vaccinated and 3 have not.(a) What is the distribution of the number of new infections among the 17 vaccinated children? What is the mean
A whooping cough outbreak. In 2007, Bob Jones University ended its fall semester a week early because of a whooping cough outbreak; 158 students were isolated and another 1200 given antibiotics as a precaution.7 Authorities react strongly to whooping cough outbreaks because the disease is so
Vaccination at work. A group of 20 children at a nursery school are exposed to whooping cough by playing with an infected child.(a) If all 20 have been vaccinated, what is the mean number of new infections?What is the probability that no more than 2 of the 20 children develop infections?(b) If none
Binomial variation. Never forget that probability describes only what happens in the long run. Example 13.5 concerns the count of bad CDs in inspection samples of size 10. The count has the binomial distribution with n = 10 and p = 0.1. The Probability applet simulates inspecting a lot of CDs if
Is this coin balanced? While he was a prisoner of war during World War II, John Kerrich tossed a coin 10,000 times. He got 5067 heads. If the coin is perfectly balanced, the probability of a head is 0.5. Is there reason to think that Kerrich’s coin was not balanced? To answer this question, find
Multiple-choice tests. Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) Answers to different
Survey demographics. According to the Census Bureau, 13% of American adults(age 18 and over) are Hispanic. An opinion poll plans to contact an SRS of 1200 adults.(a) What is the mean number of Hispanics in such samples? What is the standard deviation?(b) According to the 68–95–99.7 rule, what
High school equivalency. The Census Bureau says that 21% of Americans aged 18 to 24 do not have a high school diploma. A vocational school wants to attract young people who may enroll in order to achieve high school equivalency.The school mails an advertising flyer to 25,000 persons between the
False positives in testing for HIV. A rapid test for the presence in the blood of antibodies to HIV, the virus that causes AIDS, gives a positive result with probability about 0.004 when a person who is free of HIV antibodies is tested. A clinic tests 1000 people who are all free of HIV
Genetics. According to genetic theory, the blossom color in the second generation of a certain cross of sweet peas should be red or white in a 3:1 ratio. That is, each plant has probability 3/4 of having red blossoms, and the blossom colors of separate plants are independent.c blickwinkel/Alamy(a)
Hitting the fairway. One statistic used to assess professional golfers is driving accuracy, the percent of drives that land in the fairway. Driving accuracy for PGA Tour professionals ranges from about 40% to about 75%. Tiger Woods hits the fairway about 60% of the time.6(a) Tiger hits 14 drives in
The pill, continued. A study of the effectiveness of oral contraceptives interviews a random sample of 500 women who are taking the pill.(a) Based on the information about typical use in Exercise 13.27, what is the probability that at least 25 of these women become pregnant in the next year?(Check
On the Web, continued. A study of Internet usage interviews a random sample of 500 men aged 18 to 34. Based on the information in Exercise 13.26, what is the probability that at least 235 of the men in the sample visit an online auction site at least once a month? (Check that the Normal
The pill. Many women take oral contraceptives to prevent pregnancy. Under ideal conditions, 1% of women taking the pill become pregnant within one year. In typical use, however, 5% become pregnant.5 Choose at random 20 women taking the pill. How many become pregnant in the next year?(a) Explain why
On the Web. What kinds ofWeb sites do males aged 18 to 34 visit? About 50% of male Internet users in this age group visit an auction site such as eBay at least once a month.4 Interview a random sample of 12 male Internet users aged 18 to 34.(a) What is the distribution of the number who have
Random stock prices. A believer in the random walk theory of stock markets thinks that an index of stock prices has probability 0.65 of increasing in any year.Moreover, the change in the index in any given year is not influenced by whether it rose or fell in earlier years. Let X be the number of
Testing ESP. In a test for ESP (extrasensory perception), a subject is told that cards the experimenter can see but he cannot contain either a star, a circle, a wave, or a square. As the experimenter looks at each of 20 cards in turn, the subject names the shape on the card. A subject who is just
Binomial setting? A binomial distribution will be approximately correct as a model for one of these two sports settings and not for the other. Explain why by briefly discussing both settings.David Bergman/CORBIS(a) A National Football League kicker has made 80% of his field goal attempts in the
Binomial setting? In each situation below, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case.(a) An auto manufacturer chooses one car from each hour’s production for a detailed quality inspection. One variable recorded is the
Ten lines in the table contain 400 digits. The count of 0s in these lines is approximately Normal with(a) mean 40 and standard deviation 36.(b) mean 40 and standard deviation 6.(c) mean 36 and standard deviation 6.
The mean number of 0s in a line 40 digits long is(a) 4. (b) 3.098. (c) 0.4.
The probability of finding exactly 4 0s in a line 40 digits long is about(a) 0.0000225. (b) 0.0225. (c) 0.2059.
Abasketball player makes 70% of her free throws. She takes 7 free throws in a game.If the shots are independent of each other, the probability that she makes 5 out of the 7 shots is about(a) 0.635. (b) 0.318. (c) 0.015.Each entry in a table of random digits like Table B has probability 0.1 of being
Abasketball player makes 70% of her free throws. She takes 7 free throws in a game.If the shots are independent of each other, the probability that she makes the first 5 and misses the last 2 is about(a) 0.635. (b) 0.318. (c) 0.015.
If a basketball player makes 5 free throws and misses 2 free throws during a game, in how many ways can you arrange the sequence of hits and misses?(a)7 5= 42 (b)7 5= 21 (c)5 2= 10
In a group of 10 college students, 4 are business majors. You choose 3 of the 10 students at random and ask their major. The distribution of the number of business majors you choose is(a) binomial with n = 10 and p = 0.4.(b) binomial with n = 3 and p = 0.4.(c) not binomial.
In the previous exercise, the probability that at least 1 of Joe’s 3 eggs contains salmonella is about(a) 0.68. (b) 0.58. (c) 0.30.
Joe reads that 1 out of 4 eggs contains salmonella bacteria. So he never uses more than 3 eggs in cooking. If eggs do or don’t contain salmonella independently of each other, the number of contaminated eggs when Joe uses 3 chosen at random has the distribution(a) binomial with n = 4 and p =
Jasmine has alleles A and O. Tyrone has alleles B and O.(a) What is the probability that a child of these parents has blood type O?(b) If Jasmine and Tyrone have three children, what is the probability that all three have blood type O?(c) What is the probability that the first child has blood type
Isabel has alleles A and O. Carlos has alleles A and B. They have two children.(a) What is the probability that both children have blood type A?(b) What is the probability that both children have the same blood type?
Sarah and David both have alleles B and O.(a) What blood types can their children have?(b) What is the probability that their next child has each of these blood types?
Rachel and Jonathan both have alleles A and B.(a) What blood types can their children have?(b) What is the probability that their next child has each of these blood types?
Fundraising by telephone. Tree diagrams can organize problems having more than two stages. Figure 12.6 shows probabilities for a charity calling potential donors by telephone.12 Each person called is either a recent donor, a past donor, or a new prospect. At the next stage, the person called either
Lactose intolerance. Lactose intolerance causes difficulty digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States (ignoring other groups and people who consider themselves to belong to more than one
Where do the votes come from? In the election described in Exercise 12.49, what percent of the candidate’s votes come from black voters? (Write this as a conditional probability and use the definition of conditional probability.)
Winning at tennis, continued. Based on your work in Exercise 12.48, in what percent of points won by the server was the first serve in?(Write this as a conditional probability and use the definition of conditional probability.)
Urban voters. The voters in a large city are 40% white, 40% black, and 20% His-panic. (Hispanics may be of any race in official statistics, but here we are speaking of political blocks.) A black mayoral candidate anticipates attracting 30% of the white vote, 90% of the black vote, and 50% of the
Winning at tennis. A player serving in tennis has two chances to get a serve into play. If the first serve is out, the player serves again. If the second serve is also out, the player loses the point. Here are probabilities based on four years of theWimbledon Championship:10 P(1st serve in) = 0.59
The geometric distributions. You are tossing a pair of balanced dice in a board game. Tosses are independent. You land in a danger zone that requires you to roll doubles (both faces show the same number of spots) before you are allowed to play again. How long will you wait to play again?(a) What is
Julie’s conditional probabilities. If Julie is offered the federal job, what is the conditional probability that she is also offered the New Jersey job? If Julie is offered the New Jersey job, what is the conditional probability that she is also offered the federal job?
Will Julie get just these offers? What is the probability that Julie is offered both the Connecticut and New Jersey jobs, but not the federal job?
Will Julie get a job offer? What is the probability that Julie is offered at least one of the three jobs?
Deer and pine seedlings. In the setting of Exercise 12.41, what percent of the trees that were not damaged by deer were more than 2/3 covered by thorny plants?Julie is graduating from college. She has studied biology, chemistry, and computing and hopes to use her science background in crime
Deer and pine seedlings. In the setting of Exercise 12.41, what percent of the trees that were damaged by deer were less than 1/3 covered by thorny plants?
Deer and pine seedlings. As suburban gardeners know, deer will eat almost anything green. In a study of pine seedlings at an environmental center in Ohio, researchers noted how deer damage varied with how much of the seedling was covered by thorny undergrowth:9 Deer Damage Thorny Cover Yes No None
College degrees. Exercise 12.39 gives the counts (in thousands) of earned degrees in the United States in the 2010–2011 academic year. Use these data to answer the following questions.(a) What is the probability that a randomly chosen degree recipient is a man?(b) What is the conditional
College degrees. A striking trend in higher education is that more women than men reach each level of attainment. Here are the counts (in thousands) of earned degrees in the United States in the 2010–2011 academic year, classified by level and by the sex of the degree recipient:8 Bachelor’s
A probability teaser. Suppose (as is roughly correct) that each child born is equally likely to be a boy or a girl and that the sexes of successive children are independent.If we let BG mean that the older child is a boy and the younger child is a girl, then each of the combinations BB, BG, GB, GG
Geometric probability. Choose a point at random in the square with sides 0 ≤x ≤ 1 and 0 ≤ y ≤ 1. This means that the probability that the point falls in any region within the square is equal to the area of that region. Let X be the x coordinate and Y the y coordinate of the point chosen.
Mike’s pizza. You work at Mike’s pizza shop. You have the following information about the 7 pizzas in the oven: 3 of the 7 have thick crust, and of these one has only sausage and 2 have only mushrooms; the remaining 4 pizzas have regular crust, and of these 2 have only sausage and 2 have only
Income tax returns. Here is the distribution of the adjusted gross income (in thousands of dollars) reported on individual federal income tax returns in 2005:Income
Foreign-language study. Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of results:Language Spanish French German All others None Probability 0.26 0.09 0.03 0.03 0.59 What is the conditional probability that a
Screening job applicants.Acompany retains a psychologist to assess whether job applicants are suited for assembly-line work. The psychologist classifies applicants as one of A(well suited), B (marginal), or C (not suited). The company is concerned about the event D that an employee leaves the
Tendon surgery. You have torn a tendon and are facing surgery to repair it. The surgeon explains the risks to you: infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occur together in 1%. What percent of these operations succeed and are free from
Getting into college. Ramon has applied to both Princeton and Stanford. He thinks the probability that Princeton will admit him is 0.4, the probability that Stanford will admit him is 0.5, and the probability that both will admit him is 0.2.Make a Venn diagram. Then answer these questions.(a) What
A random walk on Wall Street? The “random walk” theory of stock prices holds that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in any one
Playing the slots. Slot machines are now video games, with outcomes determined by random number generators. In the old days, slot machines were like this: you pull the lever to spin three wheels; each wheel has 20 symbols, all equally likely to show when the wheel stops spinning; the three wheels
Universal blood donors. People with type O-negative blood are universal donors.That is, any patient can receive a transfusion of O-negative blood. Only 7.2%of the American population have O-negative blood. If 10 people appear at random to give blood, what is the probability that at least 1 of them
Playing the lottery. New York State’s “Quick Draw” lottery moves right along.Players choose between one and ten numbers from the range 1 to 80; 20 winning numbers are displayed on a screen every four minutes. If you choose just one number, your probability of winning is 20/80, or 0.25. Lester
Using the information in the previous exercise, the probability that a randomly chosen death was due to diabetes is about(a) 0.107. (b) 0.030. (c) 0.024.
Of people who died in the United States in recent years, 86% were white, 12%were black, and 2% were Asian. (This ignores a small number of deaths among other races.) Diabetes caused 2.8% of deaths among whites, 4.4% among blacks, and 3.5% among Asians. The probability that a randomly chosen death
Choose an American adult at random. The probability that you choose a woman is 0.52. The probability that the person you choose has never married is 0.25. The probability that you choose a woman who has never married is 0.11. The probability that the person you choose is either a woman or never
Let A be the event that a victim of violent death was a woman and B the event that the death was a suicide. The proportion of suicides among violent deaths of women is expressed in probability notation as(a) P(Aand B). (b) P(A | B). (c) P(B | A).
The conditional probability that the death was accidental, given that the victim was male, is about(a) 0.81. (b) 0.78. (c) 0.56.
The conditional probability that the victim was male, given that the death was accidental, is about(a) 0.81. (b) 0.78. (c) 0.56.
Choose a violent death in this age group at random. The probability that the victim was male is about(a) 0.81. (b) 0.78. (c) 0.19.
An athlete suspected of having used steroids is given two tests that operate independently of each other. Test A has probability 0.9 of being positive if steroids have been used. Test B has probability 0.8 of being positive if steroids have been used.What is the probability that neither test is
The probability that you win on one or more of your 3 plays of the game in the previous exercise is about(a) 0.02. (b) 0.06. (c) 0.999992.
An instant lottery game gives you probability 0.02 of winning on any one play. Plays are independent of each other. If you play 3 times, the probability that you win on none of your plays is about(a) 0.98. (b) 0.94. (c) 0.000008.
What’s the mean? Suppose that you roll three balanced dice. We wonder what APPLET ••• the mean number of spots on the up-faces of the three dice is. The law of large numbers says that we can find out by experience: roll three dice many times, and the average number of spots will eventually
Can we trust the central limit theorem? The central limit theorem says that“when n is large” we can act as if the distribution of a sample mean x is close to Normal. How large a sample we need depends on how far the population distribution is from being Normal. Example 11.8 shows that we can
Playing the numbers: the house has a business. Unlike Joe (see the previous exercise) the operators of the numbers racket can rely on the law of large numbers.It is said that the New York City mobster Casper Holstein took as many as 25,000 bets per day in the Prohibition era. That’s 150,000 bets
Playing the numbers: a gambler gets chance outcomes. The law of large numbers tells us what happens in the long run. Like many games of chance, the numbers racket has outcomes so variable—one three-digit number wins $600 and all others win nothing—that gamblers never reach “the long
Playing the numbers. The numbers racket is a well-entrenched illegal gambling operation in most large cities. One version works as follows: you choose one of the 1000 three-digit numbers 000 to 999 and pay your local numbers runner a dollar to enter your bet. Each day, one three-digit number is
Sampling male students, continued. To estimate the mean height μ of male students on your campus, you will measure an SRS of students. You know from government data that heights of young men are approximately Normal with standard deviation about 2.8 inches. You want your sample mean x to estimate
Sampling male students. To estimate the mean height μ of male students on your campus, you will measure an SRS of students. You know from government data that heights of young men are approximately Normal with standard deviation about 2.8 inches. How large an SRS must you take to reduce the
Airline passengers get heavier. In response to the increasing weight of airline passengers, the Federal Aviation Administration in 2003 told airlines to assume that passengers average 190 pounds in the summer, including clothing and carry-on baggage. But passengers vary, and the FAA did not specify
Returns on stocks. Andrew plans to retire in 40 years. He plans to invest part of his retirement funds in stocks, so he seeks out information on past returns. He learns that over the entire 20th century, the real (that is, adjusted for inflation) annual returns on U.S. common stocks had mean 8.7%
Pollutants in auto exhausts, continued. The level of nitrogen oxides (NOX) in the exhaust of cars of a particular model varies Normally with mean 0.2 g/mi and standard deviation 0.05 g/mi.Acompany has 25 cars of this model in its fleet. What is the level L such that the probability that the average
Auto accidents. The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. This distribution takes only whole-number values, so it is certainly not Normal.(a) Let x be the mean number of accidents per week at the intersection during a year (52
Pollutants in auto exhausts. The level of nitrogen oxides (NOX) in the exhaust of cars of a particular model varies Normally with mean 0.2 grams per mile (g/mi)and standard deviation 0.05 g/mi. Government regulations call for NOX emissions no higher than 0.3 g/mi.(a) What is the probability that a
Glucose testing, continued. Shelia’s measured glucose level one hour after a sugary drink varies according to the Normal distribution with μ = 125 mg/dl andσ = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level of 4 test results falls above L?
Durable press fabrics. “Durable press”cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also reduces the strength of the fabric. The breaking strength of untreated fabric is Normally distributed with mean 58 pounds and standard
Glucose testing. Shelia’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if
Heights of male students. To estimate the mean height μ of male students on your campus, you will measure an SRS of students. Heights of people of the same sex and similar ages are close to Normal. You know from government data that the standard deviation of the heights of young men is about 2.8
Lightning strikes. The number of lightning strikes on a square kilometer of open ground in a year has mean 6 and standard deviation 2.4. (These values are typical of much of the United States.) The National Lightning Detection Network uses automatic sensors to watch for lightning in a sample of 10
Roulette. A 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. One of the simplest wagers chooses red or black. A bet of $1 on red returns $2 if the ball lands in a red slot.
Small classes in school. The Tennessee STAR experiment randomly assigned children to regular or small classes during their first four years of school. When these children reached high school, 40.2% of blacks from small classes took the ACT or SAT college entrance exams. Only 31.7% of blacks from
Testing glass. How well materials conduct heat matters when designing houses.As a test of a new measurement process, 10 measurements are made on pieces of glass known to have conductivity 1. The average of the 10 measurements is 1.09.Is each of the boldface numbers a parameter or a statistic?
The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. The probability that the average pregnancy length for 6 randomly chosen women exceeds 270 days is about(a) 0.40. (b) 0.27. (c)
The number of hours a light bulb burns before failing varies from bulb to bulb.The distribution of burnout times is strongly skewed to the right. The central limit theorem says that(a) as we look at more and more bulbs, their average burnout time gets ever closer to the mean μ for all bulbs of
A newborn baby has extremely low birth weight (ELBW) if it weighs less than 1000 grams.Astudy of the health of such children in later years examined a random sample of 219 children. Their mean weight at birth was x = 810 grams. This sample mean is an unbiased estimator of the mean weight μ in the
Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 515 and standard deviation 114. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, the standard deviation of the average scores you get will be close to(a) 114.
Scores on the mathematics part of the SAT exam in a recent year were roughly Normal with mean 515 and standard deviation 114. You choose an SRS of 100 students and average their SAT math scores. If you do this many times, the mean of the average scores you get will be close to(a) 515. (b) 515/100 =
Annual returns on the more than 5000 common stocks available to investors vary a lot. In a recent year, the mean return was 8.3% and the standard deviation of returns was 28.5%. The law of large numbers says that(a) you can get an average return higher than the mean 8.3% by investing in a large
A study of voting chose 663 registered voters at random shortly after an election.Of these, 72% said they had voted in the election. Election records show that only 56% of registered voters voted in the election. The boldface number is a(a) sampling distribution. (b) parameter. (c) statistic.
The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 60,000 households; 4.9% of the people interviewed were unemployed. The boldface number is a(a) sampling distribution. (b) parameter. (c) statistic.
Simulating an opinion poll. An opinion poll showed that about 65% of the Amer-•••APPLET ican public have a favorable opinion of the software company Microsoft. Suppose that this is exactly true. Choosing a person at random then has probability 0.65 of getting one who has a favorable opinion
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