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inferential statistics
Stats Modeling The World AP Edition Grades 9-12 3rd Edition David E. Bock, Paul F. Velleman, Richard D. De Veaux - Solutions
Back to campus. In 2004 ACT, Inc., reported that 74%of 1644 randomly selected college freshmen returned to college the next year. Estimate the national freshman-tosophomore retention rate.a) Verify that the conditions are met.b) Construct a 98% confidence interval.c) Interpret your interval.d)
Only child. In a random survey of 226 college students, 20 reported being only children (with no siblings). Estimate the proportion of students nationwide who are only children.a) Check the conditions (to the extent you can) for constructing a confidence interval.b) Construct a 95% confidence
Pregnancy. In 1998 a San Diego reproductive clinic reported 49 live births to 207 women under the age of 40 who had previously been unable to conceive.a) Find a 90% confidence interval for the success rate at this clinic.b) Interpret your interval in this context.c) Explain what 90% confidence
Rickets. Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the 1950s, but now there is concern that a generation of children more likely
Gambling. A city ballot includes a local initiative that would legalize gambling. The issue is hotly contested, and two groups decide to conduct polls to predict the outcome.The local newspaper finds that 53% of 1200 randomly selected voters plan to vote yes, while a college Statistics class finds
Death penalty poll, part III. In the chapter s example and again in Exercise 21 we looked at a Gallup Poll investigating the public s attitude toward the death penalty. In response to one question, 60% thought it was fair, but when the question was phrased differently, the proportion in favor
Drinking. A national health organization warns that 30% of middle school students nationwide have been drunk. Concerned, a local health agency randomly and anonymously surveys 110 of the 1212 middle school students in its city. Only 21 of them report having been drunk.a) What proportion of the
Death penalty, again. In the survey on the death penalty you read about in the chapter, the Gallup Poll actually split the sample at random, asking 510 respondents the question quoted earlier, Generally speaking, do you believe the death penalty is applied fairly or unfairly in this country today?
Local news. The mayor of a small city has suggested that the state locate a new prison there, arguing that the construction project and resulting jobs will be good for the local economy. A total of 183 residents show up for a public hearing on the proposal, and a show of hands finds only 31 in
Safe food. Some food retailers propose subjecting food to a low level of radiation in order to improve safety, but sale of such irradiated food is opposed by many people. Suppose a grocer wants to find out what his customers think. He has cashiers distribute surveys at checkout and ask customers to
Junk mail. Direct mail advertisers send solicitations(a.k.a. junk mail ) to thousands of potential customers in the hope that some will buy the company s product.The acceptance rate is usually quite low. Suppose a company wants to test the response to a new flyer, and sends it to 1000 people
Teenage drivers. An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them.a) Create a 95% confidence interval for the percentage of all auto accidents that involve teenage drivers.b) Explain what your interval means.
Take the offer. First USA, a major credit card company, is planning a new offer for their current cardholders. The offer will give double airline miles on purchases for the next 6 months if the cardholder goes online and registers for the offer. To test the effectiveness of the campaign, First USA
Contributions, please. The Paralyzed Veterans of America is a philanthropic organization that relies on contributions. They send free mailing labels and greeting cards to potential donors on their list and ask for a voluntary contribution. To test a new campaign, they recently sent letters to a
Cloning 2007. A May 2007 Gallup Poll found that only 11% of a random sample of 1003 adults approved of attempts to clone a human.a) Find the margin of error for this poll if we want 95%confidence in our estimate of the percent of American adults who approve of cloning humans.b) Explain what that
Baseball fans. In a poll taken in March of 2007, Gallup asked 1006 national adults whether they were baseball fans. 36% said they were. A year previously,
Contaminated chicken, second course. The January 2007 Consumer Reports study described in Exercise 11 also found that 15% of the 525 broiler chickens tested were infected with Salmonella.a) Are the conditions for creating a confidence interval satisfied? Explain.b) Construct a 95% confidence
Contaminated chicken. In January 2007 Consumer Reports published their study of bacterial contamination of chicken sold in the United States. They purchased 525 broiler chickens from various kinds of food stores in 23 states and tested them for types of bacteria that cause food-borne illnesses.
Parole. A study of 902 decisions made by the Nebraska Board of Parole produced the following computer output.Assuming these cases are representative of all cases that may come before the Board, what can you conclude?z-Interval for proportion With 95.00% confidence, 0.56100658 , p(parole) ,
Cars. What fraction of cars is made in Japan? The computer output below summarizes the results of a random sample of 50 autos. Explain carefully what it tells you.z-Interval for proportion With 90.00% confidence, 0.29938661 , p(japan) , 0.46984416
Confidence intervals, again. Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true?a) For a given sample size, reducing the margin of error will mean lower confidence.b) For
Confidence intervals. Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true?a) For a given sample size, higher confidence means a smaller margin of error.b) For a specified
More conclusions. In January 2002, two students made worldwide headlines by spinning a Belgian euro 250 times and getting 140 heads that s 56%. That makes the 90% confidence interval (51%, 61%). What does this mean? Are these conclusions correct? Explain.a) Between 51% and 61% of all euros are
Conclusions. A catalog sales company promises to deliver orders placed on the Internet within 3 days. Followup calls to a few randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% What does this mean?Are these conclusions
More conditions. Consider each situation described.Identify the population and the sample, explain what p and represent, and tell whether the methods of this chapter can be used to create a confidence interval.a) A consumer group hoping to assess customer experiences with auto dealers surveys 167
Conditions. Consider each situation described below. Identify the population and the sample, explain what p and represent, and tell whether the methods of this chapter can be used to create a confidence interval.p*a) Police set up an auto checkpoint at which drivers are stopped and their cars
Margin of error. A medical researcher estimates the percentage of children who are exposed to lead-base paint, adding that he believes his estimate has a margin of error of about 3%. Explain what the margin of error means.
Margin of error. A TV newscaster reports the results of a poll of voters, and then says, The margin of error is plus or minus 4%. Explain carefully what that means.
Milk. Although most of us buy milk by the quart or gallon, farmers measure daily production in pounds.Ayrshire cows average 47 pounds of milk a day, with a standard deviation of 6 pounds. For Jersey cows, the mean daily production is 43 pounds, with a standard deviation of 5 pounds. Assume that
IQs. Suppose that IQs of East State University s students can be described by a Normal model with mean 130 and standard deviation 8 points. Also suppose that IQs of students from West State University can be described by a Normal model with mean 120 and standard deviation 10.a) We select a student
More groceries. Suppose the store in Exercise 50 had 312 customers today.a) Estimate the probability that the store s revenues were at least $10,000.b) If, in a typical day, the store serves 312 customers, how much does the store take in on the worst 10% of such days?
More tips. The waiter in Exercise 49 usually waits on about 40 parties over a weekend of work.a) Estimate the probability that he will earn at least $500 in tips.b) How much does he earn on the best 10% of such weekends?
Groceries. Grocery store receipts show that customer purchases have a skewed distribution with a mean of $32 and a standard deviation of $20.a) Explain why you cannot determine the probability that the next customer will spend at least $40b) Can you estimate the probability that the next 10
Tips. A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of$9.60 and a standard deviation of $5.40.a) Explain why you cannot determine the probability that a given party will tip him at least $20.b) Can you estimate the probability that the
Potato chips. The weight of potato chips in a mediumsize bag is stated to be 10 ounces. The amount that the packaging machine puts in these bags is believed to have a Normal model with mean 10.2 ounces and standard deviation 0.12 ounces.a) What fraction of all bags sold are underweight?b) Some of
Pollution. Carbon monoxide (CO) emissions for a certain kind of car vary with mean 2.9 g/mi and standard deviation 0.4 g/mi. A company has 80 of these cars in its fleet.Let represent the mean CO level for the company s fleet.a) What s the approximate model for the distribution of? Explain.b)
Joining the museum. One of the museum s phone volunteers sets a personal goal of getting an average donation of at least $100 from the new members she enrolls during the membership drive. If she gets 80 new members and they can be considered a random sample of all the museum s members, what is the
AP Stats 2006, again. An AP Statistics teacher had 63 students preparing to take the AP exam discussed in Exercise 43. Though they were obviously not a random sample, he considered his students to be typical of all the national students. What s the probability that his students will achieve an
Museum membership. A museum offers several levels of membership, as shown in the tablea) Find the mean and standard deviation of the donations.b) During their annual membership drive, they hope to sign up 50 new members each day. Would you expect the distribution of the donations for a day to
AP Stats 2006. The College Board reported the score distribution shown in the table for all students who took the 2006 AP Statistics exam.
New game. You pay $10 and roll a die. If you get a 6, you win $50. If not, you get to roll again. If you get a 6 this time, you get your $10 back.a) Create a probability model for this game.b) Find the expected value and standard deviation of your prospective winnings.c) You play this game five
Dice and dollars. You roll a die, winning nothing if the number of spots is odd, $1 for a 2 or a 4, and $10 for a 6.a) Find the expected value and standard deviation of your prospective winnings.b) You play twice. Find the mean and standard deviation of your total winnings.c) You play 40 times.
At work. Some business analysts estimate that the length of time people work at a job has a mean of 6.2 years and a standard deviation of 4.5 years.a) Explain why you suspect this distribution may be skewed to the right.b) Explain why you could estimate the probability that 100 people selected at
Pregnant again. The duration of human pregnancies may not actually follow a Normal model, as described in Exercise 37.a) Explain why it may be somewhat skewed to the left.b) If the correct model is in fact skewed, does that change your answers to partsa, b, and c of Exercise 37?Explain why or why
Rainfall. Statistics from Cornell s Northeast Regional Climate Center indicate that Ithaca, NY, gets an average of 35.4" of rain each year, with a standard deviation of 4.2". Assume that a Normal model applies.a) During what percentage of years does Ithaca get more than 40" of rain?b) Less than how
Pregnancy. Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days.a) What percentage of pregnancies should last between 270 and 280 days?b) At least how many days should the longest 25% of all pregnancies last?c) Suppose a
Safe cities. Allstate Insurance Company identified the 10 safest and 10 least-safe U.S. cities from among the 200 largest cities in the United States, based on the mean number of years drivers went between automobile accidents. The cities on both lists were all smaller than the 10 largest cities.
The trial of the pyx. In 1150, it was recognized in England that coins should have a standard weight of precious metal as the basis for their value. A guinea, for example, was supposed to contain 128 grains of gold.(There are 360 grains in an ounce.) In the trial of the pyx, coins minted under
Home values. Assessment records indicate that the value of homes in a small city is skewed right, with a mean of $140,000 and standard deviation of $60,000. To check the accuracy of the assessment data, officials plan to conduct a detailed appraisal of 100 homes selected at random. Using the 68 95
GPAs. A college s data about the incoming freshmen indicates that the mean of their high school GPAs was 3.4, with a standard deviation of 0.35; the distribution was roughly mound-shaped and only slightly skewed. The students are randomly assigned to freshman writing seminars in groups of 25. What
CEOs revisited. In Exercise 30 you looked at the annual compensation for 800 CEOs, for which the true mean and standard deviation were (in thousands of dollars) 10,307.31 and 17,964.62, respectively. A simulation drew samples of sizes 30, 50, 100, and 200 (with replacement) from the total annual
Waist size revisited. Researchers measured the Waist Sizes of 250 men in a study on body fat. The true mean and standard deviation of the Waist Sizes for the 250 men are 36.33 in and 4.019 inches, respectively. In Exercise 29 you looked at the histograms of simulations that drew samples of sizes 2,
CEO compensation. In Chapter 4 we saw the distribution of the total compensation of the chief executive officers (CEOs) of the 800 largest U.S. companies (the Fortune 800). The average compensation (in thousands of dollars) is 10,307.31 and the standard deviation is 17,964.62. Here is a histogram
Waist size. A study measured the Waist Size of 250 men, finding a mean of 36.33 inches and a standard deviation of 4.02 inches. Here is a histogram of these measurements
Sampling, part II. A sample is chosen randomly from a population that was strongly skewed to the left.a) Describe the sampling distribution model for the sample mean if the sample size is small.b) If we make the sample larger, what happens to the sampling distribution model s shape, center, and
Sampling. A sample is chosen randomly from a population that can be described by a Normal model.a) What s the sampling distribution model for the sample mean? Describe shape, center, and spread.b) If we choose a larger sample, what s the effect on this sampling distribution model?
Meals. A restauranteur anticipates serving about 180 people on a Friday evening, and believes that about 20%of the patrons will order the chef s steak special. How many of those meals should he plan on serving in order to be pretty sure of having enough steaks on hand to meet customer demand?
Nonsmokers. While some nonsmokers do not mind being seated in a smoking section of a restaurant, about 60% of the customers demand a smoke-free area. A new restaurant with 120 seats is being planned. How many seats should be in the nonsmoking area in order to be very sure of having enough seating
Genetic defect. It s believed that 4% of children have a gene that may be linked to juvenile diabetes. Researchers hoping to track 20 of these children for several years test 732 newborns for the presence of this gene. What s the probability that they find enough subjects for their study?
Apples. When a truckload of apples arrives at a packing plant, a random sample of 150 is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory. Suppose that in fact 8% of the apples on the truck do
Seeds. Information on a packet of seeds claims that the germination rate is 92%. What s the probability that more than 95% of the 160 seeds in the packet will germinate?Be sure to discuss your assumptions and check the conditions that support your model.
Polling. Just before a referendum on a school budget, a local newspaper polls 400 voters in an attempt to predict whether the budget will pass. Suppose that the budget actually has the support of 52% of the voters. What s the probability the newspaper s sample will lead them to predict defeat? Be
Binge sample. After hearing of the national result that 44% of students engage in binge drinking (5 drinks at a sitting for men, 4 for women), a professor surveyed a random sample of 244 students at his college and found that 96 of them admitted to binge drinking in the past week. Should he be
Back to school, again. Based on the 74% national retention rate described in Exercise 17, does a college where 522 of the 603 freshman returned the next year as sophomores have a right to brag that it has an unusually high retention rate? Explain.
Binge drinking. As we learned in Chapter 15, a national study found that 44% of college students engage in binge drinking (5 drinks at a sitting for men, 4 for women). Use the 68 95 99.7 Rule to describe the sampling distribution model for the proportion of students in a randomly selected group of
Back to school? Best known for its testing program, ACT, Inc., also compiles data on a variety of issues in education. In 2004 the company reported that the national college freshman-to-sophomore retention rate held steady at 74% over the previous four years. Consider colleges with freshman classes
Contacts. Assume that 30% of students at a university wear contact lenses.a) We randomly pick 100 students. Let represent the proportion of students in this sample who wear contacts. What s the appropriate model for the distribution of ? Specify the name of the distribution, the mean, and the
Loans. Based on past experience, a bank believes that 7% of the people who receive loans will not make payments on time. The bank has recently approved 200 loans.a) What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments?b) What
Mortgages. In early 2007 the Mortgage Lenders Association reported that homeowners, hit hard by rising interest rates on adjustable-rate mortgages, were defaulting in record numbers. The foreclosure rate of 1.6% meant that millions of families were in jeopardy of losing their homes. Suppose a large
Vision. It is generally believed that nearsightedness affects about 12% of all children. A school district has registered 170 incoming kindergarten children.a) Can you apply the Central Limit Theorem to describe the sampling distribution model for the sample proportion of children who are
Smoking. Public health statistics indicate that 26.4% of American adults smoke cigarettes. Using the 68 95 99.7 Rule, describe the sampling distribution model for the proportion of smokers among a randomly selected group of 50 adults. Be sure to discuss your assumptions and conditions.
Speeding. State police believe that 70% of the drivers traveling on a major interstate highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 cars.a) Using the 68 95 99.7 Rule, draw and label the distribution of the proportion of these cars the police will
Too many green ones? In a really large bag of M&M s, the students in Exercise 8 found 500 candies, and 12% of them were green. Is this an unusually large proportion of green M&M s? Explain.
Just (un)lucky? One of the students in the introductory Statistics class in Exercise 7 claims to have tossed her coin 200 times and found only 42% heads. What do you think of this claim? Explain.
Bigger bag. Suppose the class in Exercise 6 buys bigger bags of candy, with 200 M&M s each. Again the students calculate the proportion of green candies they find.a) Explain why it s appropriate to use a Normal model to describe the distribution of the proportion of green M&M s they might expect.b)
More coins. Suppose the class in Exercise 5 repeats the coin-tossing experiment.a) The students toss the coins 25 times each. Use the 68 95 99.7 Rule to describe the sampling distribution model.b) Confirm that you can use a Normal model here.c) They increase the number of tosses to 64 each. Draw
M&Ms. The candy company claims that 10% of the M&M s it produces are green. Suppose that the candies are packaged at random in small bags containing about 50 M&M s. A class of elementary school students learning about percents opens several bags, counts the various colors of the candies, and
Coin tosses. In a large class of introductory Statistics students, the professor has each person toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions.a) What
Character recognition, again. The automatic character recognition device discussed in Exercise 2 successfully reads about 85% of handwritten credit card applications.In Exercise 2 you looked at the histograms showing distributions of sample proportions from 1000 simulated samples of size 20, 50,
Send money, again. The philanthropic organization in Exercise 1 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 1 you looked at the histograms showing distributions of sample proportions from 1000 simulated mailings for samples of
Character recognition. An automatic character recognition device can successfully read about 85% of handwritten credit card applications. To estimate what might happen when this device reads a stack of applications, the company did a simulation using samples of size 20, 50, 75, and 100. For each
Send money. When they send out their fundraising letter, a philanthropic organization typically gets a return from about 5% of the people on their mailing list. To see what the response rate might be for future appeals, they did a simulation using samples of size 20, 50, 100, and 200. For each
Door prize. You are among 100 people attending a charity fundraiser at which a large-screen TV will be given away as a door prize. To determine who wins, 99 white balls and 1 red ball have been placed in a box and thoroughly mixed. The guests will line up and, one at a time, pick a ball from the
Pregnant? Suppose that 70% of the women who suspect they may be pregnant and purchase an in-home pregnancy test are actually pregnant. Further suppose that the test is 98% accurate. What s the probability that a woman whose test indicates that she is pregnant actually is?
Recalls. In a car rental company s fleet, 70% of the cars are American brands, 20% are Japanese, and the rest are German. The company notes that manufacturers recalls seem to affect 2% of the American cars, but only 1% of the others.a) What s the probability that a randomly chosen car is
The Drake equation. In 1961 astronomer Frank Drake developed an equation to try to estimate the number of extraterrestrial civilizations in our galaxy that might be able to communicate with us via radio transmissions.Now largely accepted by the scientific community, the Drake equation has helped
Coins. A coin is to be tossed 36 times.a) What are the mean and standard deviation of the number of heads?b) Suppose the resulting number of heads is unusual, two standard deviations above the mean. How many extra heads were observed?c) If the coin were tossed 100 times, would you still consider
Socks. In your sock drawer you have 4 blue socks, 5 grey socks, and 3 black ones. Half asleep one morning, you grab 2 socks at random and put them on. Find the probability you end up wearinga) 2 blue socks.b) no grey socks.c) at least 1 black sock.d) a green sock.e) matching socks.
Volcanoes. Almost every year, there is some incidence of volcanic activity on the island of Japan. In 2005 there were 5 volcanic episodes, defined as either eruptions or sizable seismic activity. Suppose the mean number of episodes is 2.4 per year. Let X be the number of episodes in the 2-year
O-rings. Failures of O-rings on the space shuttle are fairly rare, but often disastrous, events. If we are testing O-rings, suppose that the probability of a failure of any one O-ring is 0.01. Let X be the number of failures in the next 10 O-rings tested.a) What model might you use to model X?b)
Dogs. A census by the county dog control officer found that 18% of homes kept one dog as a pet, 4% had two dogs, and 1% had three or more. If a salesman visits two homes selected at random, what s the probability he encountersa) no dogs?b) some dogs?c) dogs in each home?d) more than one dog in each
Technology on campus. Every 5 years the Conference Board of the Mathematical Sciences surveys college math departments. In 2000 the board reported that 51% of all undergraduates taking Calculus I were in classes that used graphing calculators and 31% were in classes that used computer assignments.
Plan B. Here s another attempt at developing a good strategy for the dice game in Exercise 33. Instead of stopping after a certain number of rolls, you could decide to stop when your score reaches a certain number of points.a) How many points would you expect a roll to add to your score?b) In terms
When to stop? In Exercise 27 of the Review Exercises for Part III, we posed this question:You play a game that involves rolling a die. You can roll as many times as you want, and your score is the total for all the rolls. But . . . if you roll a 6, your score is 0 and your turn is over. What might
Jerseys. A Statistics professor comes home to find that all four of his children got white team shirts from soccer camp this year. He concludes that this year, unlike other years, the camp must not be using a variety of colors. But then he finds out that in each child s age group there are 4 teams,
Who s the boss? The 2000 Census revealed that 26% of all firms in the United States are owned by women. You call some firms doing business locally, assuming that the national percentage is true in your area.a) What s the probability that the first 3 you call are all owned by women?b) What s the
Teen smoking. The Centers for Disease Control say that about 30% of high-school students smoke tobacco(down from a high of 38% in 1997). Suppose you randomly select high-school students to survey them on 30. Buying melons. The first store in Exercise 28 sells watermelons for 32 cents a pound. The
Insurance. A 65-year-old woman takes out a $100,000 term life insurance policy. The company charges an annual premium of $520. Estimate the company s expected profit on such policies if mortality tables indicate that only 2.6% of women age 65 die within a year.
Stock strategy. Many investment advisors argue that after stocks have declined in value for 2 consecutive years, people should invest heavily because the market rarely declines 3 years in a row.a) Since the stock market began in 1872, there have been two consecutive losing years eight times. In six
Multiple choice. A multiple choice test has 50 questions, with 4 answer choices each. You must get at least 30 correct to pass the test, and the questions are very difficult.a) Are you likely to be able to pass by guessing on every question? Explain.b) Suppose, after studying for a while, you
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