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bayesian statistics an introduction

Statistics Through Applications 2nd Edition Daren S Starnes, David S Moore, Dan Yates - Solutions

Bias and variability Figure 5.8 (on the facing page) shows the behavior of a sample statistic in many samples in four situations. Th ese graphs are like those in Figures 5.5 and 5.6. Th at is, the heights of the bars show how oft en the sample statistic took various values in many samples from the
Dead trees On the west side of Rocky Mountain National Park, many mature pine trees are dying due to infestation by pine beetles. Scientists would like to use sampling to estimate the proportion of all pine trees in the area that have been infected.(a) Explain why it wouldn’t be practical for
Sleeping ducks, II Is the study described in the previous exercise an example of an observational study or a comparative experiment? Explain briefl y.
Sleeping ducks, I A recent report in the journal Nature examined whether ducks keep an eye out for predators while they sleep. Th e researchers, from Indiana State University, put four ducks in each of four plastic boxes, which were arranged in a row. Ducks in the two end boxes slept with one eye
Flight safety On a New York–to–Denver fl ight, 8% of the 125 passengers were selected for random security screening prior to boarding. According to the Transportation Security Administration, 10% of airline passengers are chosen for random screening.
Drink Arizona On Tuesday, the bottles of Arizona iced tea fi lled in a plant were supposed to contain an average of 20 ounces of iced tea. Quality control inspectors sampled 50 bottles at random from the day’s production. Th ese bottles contained an average of 19.6 ounces of iced tea.
Stop smoking! A random sample of 1000 people who signed a card saying they intended to quit smoking were contacted 9 months later. It turned out that 210 (21%) of the sampled individuals had not smoked over the past 6 months.Specify the population of interest, the parameter of interest, the sample,
Women count! Broadcast and print media across the country have reported on the government’s release of a report on the status of working women called“Working Women Count.” Th is report gives fi ndings from a survey of 250,000 women. According to one news report:More than 1600 businesses,
Read anything lately? Th e Denver Public Library wants to estimate the percent of Denver households with an adult who has read at least one book in the last month. Th e homes of 400 people who have library cards are sampled, and it turns out that 90% of these households have an adult who has read a
A lottery for drugs? Only limited supplies of some experimental drugs are available because the drugs are very diffi cult to make. Sometimes it is necessary to use a lottery to decide at random which patients can receive a drug that is in short supply. Several years ago, for example, Hoff man–La
Gun control Since the beginning of opinion polls over 50 years ago, at least two-thirds of those surveyed said that they favored stronger controls on fi rearms.Specifi c gun control proposals have oft en been favored by 80% to 85% of respondents.Yet little national gun control legislation has
Sampling TVs An electronics company has 50 large fl at-screen televisions ready for shipment, each labeled with one of the following serial numbers:An SRS of 5 TVs must be chosen for inspection. Use Table B to do this, beginning at line 139. Explain your method clearly. A1109 A2056 A2219 A2381
A biased sample You see a female student standing in front of the cafeteria, now and then stopping other students to ask them questions. She says that she is conducting a survey of student opinions about the quality of food in the cafeteria.Explain why this sampling method is almost certainly
More randomization Most sample surveys call residential telephone numbers at random. Th ey do not, however, always ask their questions of the person who picks up the phone. Instead, they ask about the adults who live in the residence and choose one at random to be in the sample. Why is this a good
How much do students earn? A university’s fi nancial aid offi ce wants to know how much it can expect students to earn from summer employment.Th is information will be used in setting the level of fi nancial aid. Th e population contains 3478 students who have completed at least one year of study
Random selection? Choosing at random is a “fair’’ way to decide who gets some rare prize, in the sense that everyone has the same chance to win.Random choice isn’t always a good idea—sometimes we don’t want to treat everyone the same, because some people have a better claim. In each of
Local draft boards Prior to 1970, young men were selected for military service by local draft boards. Th ere was a complex system of exemptions and quotas that allowed, for example, farmers’ sons and married young men with children to avoid the draft . Do you think that random selection among all
A call-in opinion poll Should the United Nations continue to have its headquarters in the United States? A television program asked its viewers to call in with their opinions on that question. Th ere were 186,000 callers, 67% of whom said “No.’’ A nationwide random sample of 500 adults found
Rating the police Th e Miami Police Department wants to know how black residents of Miami feel about police service. A researcher prepares several questions about the police. Th e police department chooses an SRS of 300 mailing addresses in predominantly black neighborhoods and sends a uniformed
SRS on the calculator Explain how you would use your calculator’s randInt command to perform each of the tasks in Exercise 5.13, if possible.Try each of your ideas to be sure that it works.
SRS applet Explain how you would use the Simple Random Sample applet from Activity 5.1B (page 211) to perform each of the following tasks. Try each of your ideas to be sure that it works. Can the applet be used in more than one way to carry out any of these tasks? Explain.(a) Flip a coin.(b) Shuffl
Not an SRS It’s sometimes not practical to take a simple random sample. Here’s an example that shows why. Suppose 1000 iPhones are produced at a factory today. Management would like to ensure that the phones’ display screens meet their quality control standards before shipping them to retail
Drug testing Use the table of random digits (Table B) to select an SRS of 3 of the following 25 members of an athletic team for a drug test. Be sure to say where you entered the table and how you used it. Agarwal Fuest Milhalko Shen Andrews Fuhrmann Moser Smith Baer Garcia Musselman Sundheim Berger
Is this an SRS? A university has 1000 male and 500 female faculty members. A survey of faculty opinion selects 100 of the 1000 men at random and then separately selects 50 of the 500 women at random. Th e 150 faculty members chosen make up the sample.(a) Explain why this sampling method gives each
An election day sample You want to choose an SRS of 25 of a city’s 440 voting precincts for special voting-fraud surveillance on election day.(a) Explain clearly how you would label the 440 precincts. How many digits make up each of your labels? What is the greatest number of precincts you could
How do random digits behave? Which of the following statements are true of a table of random digits, and which are false? Explain your answers.(a) Th ere are exactly four 0s in each row of 40 digits.(b) Each pair of digits has chance 1/100 of being 00.(c) Th e digits 0000 can never appear as a
Apartment living You are planning a report on apartment living in a college town. You decide to select three apartment complexes at random for in-depth interviews with residents. Use Table B, starting at line 117, to select a simple random sample of three of the following apartment complexes.
More capture-recapture Refer to Activity 5.1A (page 202). Mr. Washington’s class takes an initial sample of 50 goldfi sh. Aft er replacing the 50 tagged fi sh, the class obtains a second sample of fi sh. Of the 75 fi sh in this sample, 25 of the tagged fi sh are recaptured (along with 50 untagged
Design your own bad sample A large high school wants to gather student opinion about parking for students on campus. It isn’t practical to contact all students.(a) Give an example of a way to choose a sample of students that is bad because it depends on voluntary response.(b) Give another example
Hand-washing habits Do adults typically wash their hands aft er using the bathroom? In a telephone survey of 1001 U.S. adults, 92% said they always wash their hands aft er using a public restroom.2 An observational study of 6076 adults in public restrooms told a slightly diff erent story: only 77%
Sleepless nights How much sleep do high school students get on a typical school night? An interested student designed a survey to fi nd out. To make data collection easier, the student surveyed the fi rst 100 students to arrive at school on a particular morning. Th ese students reported an average
An online poll In June 2008, Parade magazine posed the following question:“Should drivers be banned from using all cell phones?” Readers were encouraged to vote online at parade.com. Th e July 13, 2008, issue of Parade reported the results:2407 (85%) said “Yes” and 410 (15%) said
Instant opinion A recent online poll posed the question: “Should female athletes be paid the same as men for the work they do?’’ In all, 13,147 (44%)said “Yes,’’ another 15,182 (50%) said “No,’’ and the remaining 1448 said“Don’t know.’’(a) What is the sample size for this
Teacher pay and liquor sales A study found a strong positive correlation between average teacher salaries and liquor sales over a 12-month period. Does this suggest that we should not pay teachers more because they would only spend the money on more liquor? Explain your answer clearly.
Late bloomers? From experience, Japanese cherry trees tend to blossom early when spring weather is warm and later when spring weather is cool. Here are some data on the average March temperature (in C) and the day in April when the fi rst cherry blossom appeared over a 25-year period.(a) Make a
Prediction Th e line on the scatterplot in Figure 4.32 is the least-squares regression line for predicting brain weight from body weight.(a) Suppose that a new mammal species is discovered hidden in the rain forest with body weight 600 kilograms. Predict the brain weight for this species.(b) Th e
Brain and body Th e correlation between body weight and brain weight is r = 0.86. How well does body weight explain brain weight for mammals? Give a number to answer this question, and briefl y explain what the number tells us.
Outliers Th e African elephant is much larger than any other mammal in the data set but lies roughly in the overall straight-line pattern. Dolphins, humans, and hippos lie outside the overall pattern. Th e correlation between body weight for the entire data set is r 5 0.86.(a) If we removed
Dolphins and hippos(a) Th e points for the dolphin and hippopotamus are labeled in Figure 4.32. Read from the graph the approximate body weight and brain weight for these two species.(b) One reaction to this scatterplot is “Dolphins are smart, hippos are dumb.” What feature of the plot lies
SAT preparation Can intensive preparation signifi cantly improve students’ SAT scores? Here’s one way to fi nd out. Take a random sample of 100 high school students who took the most recent SAT test and ask them how much time (in hours) they spent preparing and what score they earned on the
Why so small?(a) Make a scatterplot of the following data:(b) Use the method of Example 4.5 (page 158) to show that the correlation is about 0.5.(c) What feature of the data is responsible for reducing the correlation to this value despite a strong straight-line association between x and y in most
When it rains, it pours Figure 4.31 plots the highest yearly precipitation ever recorded in each state against the highest daily precipitation ever recorded in that state. Th e points for Alaska, Hawaii, and Texas are marked on the scatterplot.(a) About what are the highest daily and yearly
Measuring crickets For a biology project, you measure the length (centimeters)and weight (grams) of 12 crickets.(a) Explain why you expect the correlation between length and weight to be positive.(b) If you measured length in inches, how would the correlation change? (Th ere are 2.54 centimeters in
Calculating the least-squares line Do you like to know the mathematical details when you study something? Here is the formula for the least-squares regression line for predicting y from x. Start with the means and the standard deviations of the two variables and the correlation r between them. Th e
Miscarriages among workers A study showed that women who work in the production of computer chips have abnormally high numbers of miscarriages.Th e union claimed that exposure to chemicals used in production caused the miscarriages.Another possible explanation is that these workers spend most of
More chirping crickets In late summer 2007, Dr. Peggy LeMone decided to collect some cricket chirp and temperature data at her home in Colorado.She made several measurements of the number of cricket chirps in a 15-second period and the outdoor temperature in degrees Fahrenheit.20 Figure 4.29 shows
Prediction?(a) Use the regression equation in the previous exercise to predict the weight of the soap aft er 30 days. What’s wrong with using the regression line to predict weight aft er 30 days?(b) For these data, r2 0.996. Explain what this value means.(c) Make a residual plot for these data.
Regression Th e equation for the least-squares regression line for the data in Table 4.4 is weight 133.2 6.31(day)(a) Explain carefully what the slope b 6.31 tells us about how fast the soap lost weight.(b) Interpret the y intercept, 133.2.(c) Mr. Boggs did not measure the weight of the soap
Scatterplot Plot the weight of the bar of soap against day. Is the overall pattern roughly a straight line? Based on your scatterplot, is the correlation between day and weight close to 1, positive but not close to 1, close to 0, negative but not close to 1, or close to 1? Explain your answer.
Application 4.2 follow-up Refer to Application 4.2 (page 189).(a) Suppose we want to predict the number of cricket chirps per second when the outdoor temperature is 80F. Should we use the regression equation from Application==4.2? Why or why not?(b) If we changed the temperature data in
Obesity in mothers and daughters A study found that the correlation between the body mass index (BMI) of young girls and their minutes of physical activity in a day was r 520.18.19 Why might we expect this correlation to be negative? What percent of the variation in BMI among the girls in the study
Beer drinking and cancer A study using data from 41 states found a positive correlation between beer consumption per person and death rates from some forms of cancer. Th e states with the highest death rates from these types of cancer were Rhode Island and New York. Th e beer consumption in those
Heart attacks If you need medical care, should you go to a hospital that handles many cases like yours? Figure 4.28 presents some data for heart attacks. Th e fi gure plots mortality (the proportion of patients who died) against the number of heart attack patients treated for a large number of
Predicting Old Faithful eruptions Figure 4.27 shows the results from a least-squares regression on the Old Faithful eruption data from the beginning of the chapter. Length of previous eruption is the explanatory variable, and time until the next eruption is the response variable. Both variables
Is math the key to success in college? Here is the opening of a newspaper account of a College Board study of 15,941 high school graduates:Minority students who take high school algebra and geometry succeed in college at almost the same rate as whites, a new study says. Th e link between high
Snow and earthquakes A study measures the average annual snowfall (in inches) for 10 cities over the last decade along with the greatest earth movement(on the Richter scale) over this same time period. Th e study included 5 cities in California’s San Francisco Bay Area and 5 cities from
Education and income Th ere is a strong positive correlation between years of schooling completed and lifetime earnings for American men. One possible reason for this association is causation: more education leads to higher-paying jobs. Another explanation is confounding: men who complete many
Is a little alcohol good for you? A survey of 7000 California men found little correlation between alcohol consumption and chance of dying during the 51⁄2 years of the study. In fact, men who did not drink at all during these years had a slightly higher death rate than did light drinkers. Th is
Ice cream and drowning Th ere is a positive association between ice cream sales and the number of people who drown. Does that mean eating ice cream causes people to drown? Of course not. Give a more plausible explanation that involves common response. Draw a picture like Figure 4.26 (page 186) to
Beer and BAC, II Refer to the previous exercise.(a) If a student drinks 5 beers, on average what do you predict the student’s BAC will be? Show your work.(b) Would the regression method be as accurate for predicting BAC for a person who drinks 15 beers? Explain.(c) Use your calculator to
Beer and BAC, I How much does drinking beer increase the alcohol content of your blood? Th is question was addressed in an experiment at Ohio State University. Sixteen students volunteered to participate. Before the experiment, each student blew into a Breathalyzer machine to show that their blood
r versus r2 “When r 5 0.7, this means that y can be predicted from x for 70% of the individuals in the sample.” Is this statement true or false? Would it be true if r2 5 0.7? Explain your answers.4.36 Correlation and regression If the correlation between two variables x and y is r 5 0, there is
Joan’s gas use residuals In the Calculator Corner, we constructed a residual plot for Joan’s gas consumption data. Confi rm the value of the residual shown in the TI-84 screen shot for the month with an average temperature of 49.4°F. Show your work.
Hiker residuals Here, one more time, are the data on body and backpack weights of the 8 hikers: Body weight (lb): 120 187 109 103 131 165 Backpack weight (lb): 26 30 26 24 29 555 158 116 35 31 28
Sleep debt A researcher reported that the average teenager needs 9.3 hours of sleep per night but gets only 6.3 hours of sleep per night.12 By the end of a 5-day school week, a teenager would accumulate about 15 hours of “sleep debt.” Students in a high school statistics class were skeptical,
Least-squares idea Th e TI-Nspire screen shot below shows a scatterplot of fi ve data points. Which of the following two lines “fi ts”the data better: y 5 1 2 x or y 5 3 2 2x? Justify your answer by computing the sum of the squares of the vertical distances of each point from the two lines.
Least-squares pizza Copy Figure 4.20 onto your paper. Use the fi gure to help you explain the meaning of “least-squares regression line.”
Pizza party prediction Two players, John and Ezekiel, arrived late for the party.John ate 8 slices of pizza. Ezekiel showed off by eating 16 slices.(a) Use the regression line to predict the number of laps that John will complete.(b) Can you use the regression line to predict how many laps Ezekiel
Predicting gas usage In Example 4.7 (page 170), we examined the relationship between the average monthly temperature and the amount of natural gas consumed in Joan’s midwestern home. Th e equation of the least-squares line shown in Figure 4.18 is y 5 1425 2 19.87x.(a) Identify the slope of the
Strong relationship, low correlation Make a scatterplot that shows a strong relationship between two variables but a low correlation.
Foot problems Metatarsus adductus (call it MA) is a turning in of the front part of the foot that is common in adolescents and usually corrects itself.Hallux abducto valgus (call it HAV) is a deformation of the big toe that is not common in youth and oft en requires surgery. Perhaps the severity of
Guess the correlation For each of the following pairs of variables, would you expect a large negative correlation, a large positive correlation, or a small correlation? Justify your answers.(a) Th e age of secondhand cars and their prices(b) Th e weight of new cars and their gas mileages in miles
Pollution from vehicle exhaust Auto manufacturers are required to test their vehicles for the amount of each of several pollutants in the exhaust. Th e amount of a pollutant varies even among identical vehicles, so that several vehicles must be tested. Figure 4.16 is a scatterplot of the amounts of
Correlation “facts” Which of these statements are true and which are false?(a) A correlation of 0.8 means that 80% of the points in the scatterplot lie on a line.(b) If the correlation between two variables is zero, then there can be no relationship between them.(c) For all books in the Library
What number can I be?(a) What are all the values that a correlation r can possibly take?(b) What are all the values that a standard deviation s can possibly take?
IQ and siblings Th e correlation between a child’s score on the vocabulary portion of the Wechsler Intelligence Scale for Children (a standard IQ test) and the number of siblings a child has is r 0.319.9(a) Explain in words what this r says.(b) Can you suggest an explanation for this
Born to be old? Is there a relationship between the gestational period (time from conception to birth) of an animal and its average life span? Figure 4.15 is a scatterplot of the gestational period and average life span for 43 species of animals.(a) Describe the direction, form, and strength of the
NAEP and poverty Refer to the previous exercise. What is the relationship between the percent of students in a state who were eligible for free or reduced-price school lunches and the percent who scored profi cient on the NAEP mathematics test? Support your answer with a well-labeled
NAEP for fourth-graders Th e National Assessment of Educational Progress (NAEP) assesses what students know in several subject areas based on large representative samples. Table 4.2 (on the facing page) reports some fi ndings of the NAEP year 2007 Mathematics Assessment for fourth-graders in the
Teen drivers Th irty students in a high school statistics class were asked to report the age (in years) and odometer reading of their primary vehicle. Here are their data:(a) Enter the data into your calculator lists. Refer to the Calculator Corner (page 152).(b) Make a scatterplot of the data.
Interested in stocks? When interest rates are high, investors may avoid stocks because they can get high returns with less risk. Figure 4.9 plots the annual returns on U.S. common stocks over a 50-year period against the returns on Treasury bills for the same years.5 (Th e interest paid by Treasury
Crawl before you walk At what age do babies learn to crawl? Does it take longer to learn in the winter, when babies are oft en bundled in clothes that restrict their movement? Perhaps there might even be an association between babies’ crawling age and the average temperature during the month they
Rich states, poor states One measure of a state’s prosperity is the median income of its households. Another measure is the mean personal income per person in the state. Figure 4.8 is a scatterplot of these two variables, both measured in dollars.(a) Explain why you expect a positive association
Fast cars Interested in a sporty car? Worried that it might use too much gas? Th e Environmental Protection Agency lists most such vehicles in its“minicompact” or “two-seater” categories. Table 4.1 gives city and highway gas mileages (in miles per gallon) for all model year 2009 cars in
Healthy breeding Oft en the percent of an animal species in the wild that survive to breed again is lower following a successful breeding season.Th is is part of nature’s self-regulation, tending to keep population size stable.A study of merlins (small falcons) in northern Sweden observed the
SAT scores by state More than 1.5 million students took the SAT during the 2007–2008 school year. For the entire group of test takers, the mean score on the Math test was 515. If we look at the data on a state-by-state basis, we see quite a diff erent picture. Figure 4.5 is a scatterplot of the
IQ and GPA Figure 4.4 is a scatterplot of school grade point average versus IQ score for all 78 seventh-grade students in a rural midwestern school.(a) Describe the overall pattern of the relationship in words. (Points A, B, and C might be called outliers.)(b) About what are the IQ and GPA for
Explanatory or response, II In each of the following situations, is it more reasonable to simply explore the relationship between the two variables or to view one of the variables as an explanatory variable and the other as a response variable? In the latter case, which is the explanatory variable
Explanatory or response, I In each of the following situations, is it more reasonable to simply explore the relationship between the two variables or to view one of the variables as an explanatory variable and the other as a response variable? In the latter case, which is the explanatory variable
Do you sudoku? In the chapter opening story (page 101), one of the authors played an online game of sudoku. At the end of his game, this graph was displayed.Th e density curve shown was constructed from a histogram of times from 4,000,000 games played in one week at this Web site.(a) How would you
Th e stock market Th e annual rate of return on stock indexes (which combine many individual stocks) is very roughly Normal. Since 1945, the Standard & Poor’s 500 index has had a mean yearly return of 12%, with a standard deviation of 16.5%. Take this Normal distribution to be the distribution of
Helmet sizes Th e army reports that the distribution of head circumference among soldiers is approximately Normal with mean 22.8 inches and standard deviation 1.1 inches. Helmets are mass-produced for all except the smallest 5% and the largest 5% of head sizes. Soldiers in the smallest or largest
Density curves Figure 3.27 (on the facing page) shows density curves of two diff erent shapes. Briefl y describe the overall shape of each distribution. Two or three points are marked on each curve. Th e mean and the median are among these points.For each curve, which point is the median and which
Bacteria in milk A study of bacterial contamination in milk counted the number of coliform bacteria per milliliter in 100 specimens of milk purchased in East Coast grocery stores. Th e U.S. Public Health Service recommends no more than 10 coliform bacteria per milliliter. Here are the data:(a)
Low-birth-weight babies Researchers in Norway analyzed data on the birth weights of 400,000 newborns over a 6-year period. Th e distribution of birth weights is approximately Normal with a mean of 3668 grams and a standard deviation of 511 grams.6 Babies that weigh less than 2500 grams at birth are
Finding z-scores Use Table A to fi nd the value z from a standard Normal distribution that satisfi es each of the following conditions. In each case, sketch a standard Normal curve with your value of z marked on the axis. Use an applet or your calculator to check your answers.(a) Th e point z with
Finding areas Use Table A to fi nd the proportion of observations from a standard Normal distribution that falls in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region. Use an applet or your calculator to check your answers.(a) z $
Standardized test scores as percentiles Joey received a report that he scored in the 97th percentile on a national standardized math test but in the 72nd percentile on the reading portion of the test.(a) Explain to Joey’s grandmother, who knows little statistics, what these numbers mean.(b) Can
Textbook costs Students taking a college introductory statistics class reported spending an average of $305 on textbooks that quarter with a standard deviation of $90. Here’s a rough sketch of a Normal density curve that fi tted the histogram well:(a) Approximately what percent of the students
Get smart Refer to the previous exercise. What IQ score was at the 98th percentile in 1932? Show your work.

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