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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
Lunchtime. Create and interpret a model for the toddlers lunchtime data presented in Chapter
Twins. Twins are often born after a pregnancy that lasts less than 9 months. The graph from the Journal of the American Medical Association (JAMA) shows the rate of preterm twin births in the United States over the past 20 years. In this study, JAMA categorized mothers by the level of prenatal
French. Consider the association between a student s score on a French vocabulary test and the weight of the student. What direction and strength of correlation would you expect in each of the following situations? Explain.a) The students are all in third grade.
Modeling jumps. Here are the summary statistics for the Olympic long jumps and high jumps displayed in the scatterplot above:a) Write the equation of the line of regression for estimating High Jump from Long Jump.b) Interpret the slope of the line.c) In a year when the long jump is 350 inches, what
Jumps 2004. How are Olympic performances in various events related? The plot shows winning long-jump and high-jump distances, in inches, for the Summer Olympics from 1912 through 2004.a) Describe the association.b) Do long-jump performances somehow influence the high-jumpers? How do you account for
Depression. The September 1998 issue of the American Psychologist published an article by Kraut et al. that reported on an experiment examining the social and psychological impact of the Internet on 169 people in 73 households during their first 1 to 2 years online. In the experiment, 73 households
Winter in the city. Summary statistics for the data relating the latitude and average January temperature for 55 large U.S. cities are given below.Variable Mean StdDev Latitude 39.02 5.42 JanTemp 26.44 13.49 Correlation + *0.848a) What percent of the variation in January Temperatures can be
Correlations. The study of U.S. cities in Exercise 25 found the mean January Temperature (degrees Fahrenheit), Altitude (feet above sea level), and Latitude (degrees north of the equator) for 55 cities. Here s the correlation matrix:
US Cities. Data from 50 large U.S. cities show the mean January Temperature and the Latitude. Describe what you see in the scatterplot.
Tips. It s commonly believed that people use tips to reward good service. A researcher for the hospitality industry examined tips and ratings of service quality from 2645 dining parties at 21 different restaurants. The correlation between ratings of service and tip percentages was 0.11. (M. Lynn
No smoking? The downward trend in smoking you saw in the last exercise is good news for the health of babies, but will it ever stop?a) Explain why you can t use the linear model you created in Exercise 22 to see when smoking during pregnancy will cease altogether.b) Create a model that could
New homes. A real estate agent collects data to develop a model that will use the Size of a new home (in square feet) to predict its Sale Price (in thousands of dollars). Which of these is most likely to be the slope of the regression line: 0.008, 0.08, 0.8, or 8? Explain.a) Create a scatterplot
Improving trees. In the last exercise you saw that the linear model had some deficiencies. Let s create a better model.a) Perhaps the cross-sectional area of a tree would be a better predictor of its age. Since area is measured in square units, try re-expressing the data by squaring the diameters.
How old is that tree? One can determine how old a tree is by counting its rings, but that requires cutting the tree down. Can we estimate the tree s age simply from its diameter? A forester measured 27 trees of the same species that had been cut down, and counted the rings to determine the ages of
Which croc? The ranges inhabited by the Indian gharial crocodile and the Australian saltwater crocodile overlap in Bangladesh. Suppose a very large crocodile skeleton is found there, and we wish to determine the species of the animal. Wildlife scientists have measured the lengths of the heads and
Old Faithful. There is evidence that eruptions of Old Faithful can best be predicted by knowing the duration of the previous eruption.a. Describe what you see in the scatterplot of Intervals between eruptions vs. Duration of the previous eruption
Colorblind. Although some women are colorblind, this condition is found primarily in men. Why is it wrong to say there s a strong correlation between Sex and Colorblindness?
Cars, one more time! Can we predict the Horsepower of the engine that manufacturers will put in a car by knowing the Weight of the ca?
Autos revisited. Look again at the correlation table for cars in the previous exercise.a) Which two variables in the table exhibit the strongest association?b) Is that strong association necessarily cause-and-effect?Offer at least two explanations why that association might be so strong.c) Engine
Correlations. What factor most explains differences in Fuel Efficiency among cars? On the next page is a correlation matrix exploring that relationship for the car s Weight, Horsepower, engine size (Displacement), and number of Cylinders.
Cramming. One Thursday, researchers gave students enrolled in a section of basic Spanish a set of 50 new vocabulary words to memorize. On Friday the students took a vocabulary test. When they returned to class the following Monday, they were retested without advance warning. Both sets of test
Traffic. Highway planners investigated the relationship between traffic Density (number of automobiles per mile)and the average Speed of the traffic on a moderately large city thoroughfare. The data were collected at the same location at 10 different times over a span of 3 months.They found a mean
Grades. A Statistics instructor created a linear regression equation to predict students final exam scores from their midterm exam scores. The regression equation was Mid.a) If Susan scored a 70 on the midterm, what did the instructor predict for her score on the final?b) Susan got an 80 on the
A manatee model 2005. Continue your analysis of the manatee situation from the previous exercise.a) Create a linear model of the association between Manatee Deaths and Powerboat Registrations.b) Interpret the slope of your model.c) Interpret the y-intercept of your model.d) How accurately did your
Manatees 2005. Marine biologists warn that the growing number of powerboats registered in Florida threatens the existence of manatees. The data below come from the Florida Fish and Wildlife Conservation Commission(www.floridamarine.org) and the National Marine Manufacturers Association
Acid rain. Biologists studying the effects of acid rain on wildlife collected data from 163 streams in the Adirondack Mountains. They recorded the pH (acidity) of the water and the BCI, a measure of biological diversity, anda) What is the correlation between pH and BCI?b) Describe the association
Dow Jones 2006. The Dow Jones stock index measures the performance of the stocks of America s largest companies (http://finance.yahoo.com). A regression of the Dow prices on years 1972 2006 looks like this:
More twins 2004? As the table on the next page shows, the number of twins born in the United States has been increasing. (www.cdc.gov/nchs/births.htm)a) Find the equation of the regression line for predicting the number of twin births.b) Explain in this context what the slope of this line means.c)
Vineyards again. Instead of Age, perhaps the Size of the vineyard (in acres) is associated with the price of the wines. Look at the scatterplot:a) Do you see any evidence of an association?b) What concern do you have about this scatterplot?c) If the red data point is removed, would you expect the
Vineyards. Shown below are the scatterplot and regression analysis for Case Prices of 36 wines from vineyards in the Finger Lakes region of New York State and the Ages of the vineyards.a) Does it appear that vineyards in business longer get higher prices for their wines? Explain.b) What does this
Togetherness. Are good grades in high school associated with family togetherness? A random sample of 142 high school students was asked how many meals per week their families ate together. Their responses produced a mean of 3.78 meals per week, with a standard deviation of 2.2. Researchers then
College. Every year US News and World Report publishes a special issue on many U.S. colleges and universities. The scatterplots below have Student/Faculty Ratio(number of students per faculty member) for the colleges and universities on the y-axes plotted against 4 other variables. The correct
Human Development Index revisited. In Exercise 35 we examined the relationship between log(GDPPC) and HDI for 172 countries. The number of cell phone subscribers (per 1000 people) is also positively associated with economic progress in a country. Here s a scatterplot of CellPhones (subscribers per
Human Development Index. In Exercise 3 of Chapter 9 we saw that the United Nations Development Programme (UNDP) uses the Human Development Index(HDI) in an attempt to summarize the progress in health, education, and economics of a country with one number.The gross domestic product per capita
Oil production 2005 (again). In the next column are the data on U.S. oil production first seen in Exercise 42 of Chapter 7.How successfully might a model based on these data predict the future of U.S. oil production? Explain.
Internet. It s often difficult to find the ideal model for situations in which the data are strongly curved. The table below shows the rapid growth of the number of academic joua) Try to create a good model to describe this growth.b) Use your model to estimate the number of electronic journals in
Tree growth. A 1996 study examined the growth of grapefruit trees in Texas, determining the average trunk Diameter (in inches) for trees of varying Ages:a) Fit a linear model to these data. What concerns do you have about the model?b) If data had been given for individual trees instead of averages,
Years to live 2003. Insurance companies and other organizations use actuarial tables to estimate the remaining life-spans of their customers. On the next page are the estimated additional years of life for black males in the United States, according to a 2003 National Vital Statistics Report.
Orange production. The table below shows that as the number of oranges on a tree increases, the fruit tends to get smaller. Create a model for this relationship, and express any concerns you may have.
Slower is cheaper? Researchers studying how a car s Fuel Efficiency varies with its Speed drove a compact car 200 miles at various speeds on a test track. Their data are shown in the table
Lifting more weight 2004. In Exercise 26 you examined the winning weight-lifting performances for the 2004 Olympics. One of the competitors turned in a performance that appears not to fit the model you created.a) Consider that competitor to be an outlier. Eliminate that data point and re-create
Life expectancy. The data in the next column list the Life Expectancy for white males in the United States every decade during the last century ( to 1910, to 1920, etc.). Create a model to predict future increases in life expectancy. (National Vital Statistics Report)
Weightlifting 2004. Listed below are the gold medal winning men s weight-lifting performances at the 2004 Olympics.a) Create a linear model for the Weight Lifted in each Weight Class.b) Check the residuals plot. Is your linear model appropriate?c) Create a better model.d) Explain why you think your
Logs (not logarithms). The value of a log is based on the number of board feet of lumber the log may contain.(A board foot is the equivalent of a piece of wood 1 inch thick, 12 inches wide, and 1 foot long. For example, a piece that is 12 feet long contains 8 board feet.)To estimate the amount of
Models and laws: Planets 2006 part 5. The model you found in Exercise 20 is a relationship noted in the 17th century by Kepler as his Third Law of Planetary Motion. It was subsequently explained as a consequence of Newton s Law of Gravitation. The models for Exercises 21 23 relate to what is
Eris: Planets 2006, part 4. In July 2005, astronomers Mike Brown, Chad Trujillo, and David Rabinowitz announced the discovery of a sun-orbiting object, since named Eris,6 that is 5% larger than Pluto. Eris orbits the sun once every 560 earth years at an average distance of about 6300 million miles
Planets 2006, part 3. The asteroid belt between Mars and Jupiter may be the remnants of a failed planet. If so, then Jupiter is really in position 6, Saturn is in 7, and so on. Repeat Exercise 21, using this revised method of numbering the positions. Which method seems to work better?
Planet distances and order 2006. Let s look again at the pattern in the locations of the planets in our solar system seen in the table in Exercise 20.a) Re-express the distances to create a model for the Distance from the sun based on the planet s Position.b) Based on this model, would you agree
Planet distances and years 2006. At a meeting of the International Astronomical Union (IAU) in Prague in 2006, Pluto was determined not to be a planet, but rather the largest member of the Kuiper belt of icy objects. Let s examine some facts. In the next column is a table of the 9 sun-orbiting
Baseball salaries 2005. Ballplayers have been signing ever larger contracts. The highest salaries (in millions of dollars per season) for some notable players are given in the following table.a) Examine a scatterplot of the data. Does it look straight?b) Find the regression of Salary vs. Year and
Pendulum. A student experimenting with a pendulum counted the number of full swings the pendulum made in 20 seconds for various lengths of string. Her data are shown on the next page.
Brakes. The table below shows stopping distances in feet for a car tested 3 times at each of 5 speeds. We hope to create a model that predicts Stopping Distance from the Speed of the car.a) Explain why a linear model is not appropriate.b) Re-express the data to straighten the scatterplot.c) Create
Pressure. Scientist Robert Boyle examined the relationship between the volume in which a gas is contained and the pressure in its container. He used a cylindrical container with a moveable top that could be raised or lowered to change the volume. He measured the Height in inches by counting equally
Here are a regression and residual plot when we use the log of GDP in the model. Is this a better model for GDP? Explain.
Better GDP model? Consider again the post-1950 trend in U.S. GDP we examined in Exercise
Catapults. Ancient torsion catapults were machines for throwing rocks or other projectiles. These catapults used a pair of torsion springs made of a bundle of horsehair rope to accelerate the projectile. Greek engineers used reexpressions. They determined that they could design catapults of almost
GDP. The scatterplot shows the gross domestic product(GDP) of the United States in billions of dollars plotted against years since 1950.
Treasury Bills. The 3-month Treasury bill interest rate is watched by investors and economists. Here s a scatterplot of the 3-month Treasury bill rate since 1950:
Earnings/price ratio. Investors often judge the value of a stock by considering the company s earnings (after taxes) relative to the stock price. If the ratio is high, then the stock may be a good buy. Here s a scatterplot of the one stock s earnings/price ratio since 1950:
Crowdedness again. In Exercise 8 we looked at United Nations data about a country s GDP and the average number of people per room (Crowdedness) in housing there. For a re-expression, a student tried the reciprocal 210000/GDP, representing the number of people per$10,000 of gross domestic product.
Here are the revised regression analysis and residuals plot:
Gas mileage revisited. Let s try the re-expressed variable Fuel Consumption (gal/100 mi) to examine the fuel efficiency of the 11 cars in Exercise
Crowdedness. In a Chance magazine article (Summer 2005), Danielle Vasilescu and Howard Wainer used data from the United Nations Center for Human Settlements to investigate aspects of living conditions for several countries. Among the variables they looked at were the country s per capita gross
Gas mileage. As the example in the chapter indicates, one of the important factors determining a car s Fuel Efficiency is its Weight. Let s examine this relationship again, for 11 cars.a) Describe the association between these variables shown in the scatterplot.
More models. For each of the models listed below, predict y when .a)b)c)d)
Models. For each of the models listed below, predict y when .a)b)c)d)e)
Airline passengers revisited. In Chapter 9, Exercise 9, we created a linear model describing the trend in the number of passengers departing from Oakland (CA)airport each month since the start of 1997. Here s the residual plot, but with lines added to show the order of the values in time:a) Can you
Residuals. Suppose you have fit a linear model to some data and now take a look at the residuals. For each of the following possible residuals plots, tell whether you would try a re-expression and, if so, why.
Residuals. Suppose you have fit a linear model to some data and now take a look at the residuals. For each of the following possible residuals plots, tell whether you would try a re-expression and, if so, why.a)b) c)
Second stage 2007. Look once more at the data from the Tour de France. In Exercise 32 we looked at the whole history of the race, but now let s consider just the post World War II era.a) Find the regression of Avg Speed by Year only for years from 1947 to the present. Are the conditions for
Inflation 2006. The Consumer Price Index (CPI) tracks the prices of consumer goods in the United States, as shown in the table (ftp://ftp.bis.gov). It indicates, for example, that the average item costing $17.70 in 1926 cost$201.60 in the year 2006.
Tour de France 2007. We met the Tour de France data set in Chapter 2 (in Just Checking). One hundred years ago, the fastest rider finished the course at an average speed of about 25.3 kph (around 15.8 mph). In 2005, Lance Armstrong averaged 41.65 kph (25.88 mph) for the fastest average winning
Life Expectancy 2004. Data from the World Bank for 26 Western Hemisphere countries can be used to examine the association between female Life Expectancy and the average Number of Children women give birth to.(http://devdata.worldbank.org/data-query/)
Unwed births. The National Center for Health Statistics reported the data below, showing the percentage of all births that are to unmarried women for selected years between 1980 and 1998. Create a model that describes this trend. Justify decisions you make about how to best use these data.
Marriage age 2003 revisited. Suppose you wanted to predict the trend in marriage age for American women into the early part of this century.a) How could you use the data graphed in Exercise 1 to get a good prediction? Marriage ages in selected years starting in 1900 are listed below. Use all or
Another swim 2006. In Exercise 26 we saw that Vicki Keith s round-trip swim of Lake Ontario was an obvious outlier among the other one-way times. Here is the new regression after this unusual point is removed:Dependent variable is: Time R-squared * 4.1% s * 292.6 Variable Coefficient Intercept
Elephants and hippos. We removed humans from the scatterplot in Exercise 25 because our species was an outlier in life expectancy. The resulting scatterplot shows two points that now may be of concern. The point in the upper right corner of this scatterplot is for elephants, and the other point at
Swim the lake 2006. People swam across Lake Ontario 42 times between 1974 and 2006 (www.soloswims.com).We might be interested in whether they are getting any faster or slower. Here are the regression of the crossing Times (minutes) against the Year of the crossing and the residuals plota) What does
Gestation. For women, pregnancy lasts about 9 months. In other species of animals, the length of time from conception to birth varies. Is there any evidence that the gestation period is related to the animal s lifespan?The first scatterplot shows Gestation Period (in days) vs.Life Expectancy (in
Ages of couples, again. Has the trend of decreasing difference in age at first marriage seen in Exercise 22 gotten stronger recently? Here are the scatterplot and residual plot for the data from 1975 through 2003, along with a regression for just those years.
Interest rates revisited. In Exercise 21 you investigated the federal rate on 3-month Treasury bills between 1950 and 1980. The scatterplot below shows that the trend changed dramatically after 1980, so we ve built a new regression model for the data since 1980a) How does this model compare to the
Ages of couples 2003. The graph shows the ages of both men and women at first marriage. (www.census. gov)Clearly, the pattern for men is similar to the pattern for women. But are the two lines getting closer together?Here s a timeplot showing the difference in average age (men s age 2 women s age)
Interest rates. Here s a plot showing the federal rate on 3-month Treasury bills from 1950 to 1980, and a regres sion model fit to the relationship between the Rate (in %)and Years since 1950. (www.gpoaccess.gov/eop/)
Speed. How does the speed at which you drive affect your fuel economy? To find out, researchers drove a compact car for 200 miles at speeds ranging from 35 to 75 miles per hour. From their data, they created the model Fuel Efficiency * 32 + 0.1 Speed and created this residual plot:
Heating. After keeping track of his heating expenses for several winters, a homeowner believes he can estimate the monthly cost from the average daily Fahrenheit temperature by using the model Cost * 133 + 2.13 Temp.The residuals plot for his data is shown.
Grades. A college admissions officer, defending the college s use of SAT scores in the admissions process, produced the graph below. It shows the mean GPAs for last year s freshmen, grouped by SAT scores. How strong is the evidence that SAT Score is a good predictor of GPA?What concerns you about
Reading. To measure progress in reading ability, students at an elementary school take a reading comprehension test every year. Scores are measured in grade-level units; that is, a score of 4.2 means that a student is reading at slightly above the expected level for a fourth grader. The school
What s the effect? A researcher studying violent behavior in elementary-school children asks the children s parents how much time each child spends playing computer games and has their teachers rate each child on the level of aggressiveness they display while playing with other children. Suppose
What s the cause? Suppose a researcher studying health issues measures blood pressure and the percentage of body fat for several adult males and finds a strong positive association. Describe three different possible cause-and-effect relationships that might be present.
The extra point revisited. The original five points in Exercise 13 produce a regression line with slope 0. Match each of the green points (ae) with the slope of the line after that one point is added:1)2)3) 0.00 4) 0.05 5) 0.85
The extra point. The scatterplot shows five blue data points at the left. Not surprisingly, the correlation for these points is . Suppose one additional data point is added at one of the five positions suggested below in green. Match each point (ae) with the correct new correlation from the list
More unusual points. Each of the following scatterplots shows a cluster of points and one stray point. For each, answer these questions:1) In what way is the point unusual? Does it have high leverage, a large residual, or both?2) Do you think that point is an influential point?3) If that point were
Unusual points. Each of the four scatterplots that follow shows a cluster of points and one stray point. For each, answer these questions:1) In what way is the point unusual? Does it have high leverage, a large residual, or both?2) Do you think that point is an influential point?3) If that point
Tracking hurricanes. In a previous chapter, we saw data on the errors (in nautical miles) made by the National Hurricane Center in predicting the path of hurricanes. The following scatterplot shows the trend in the 24-hour tracking errors since 1970. (www.nhc.noaa.gov)a) Interpret the slope and
Oakland passengers. The scatterplot below shows the number of passengers departing from Oakland (CA)airport month by month since the start of 1997. Time is shown as years since 1990, with fractional years used to represent each month. (Thus, June of 1997 is 7.5 halfway through the 7th year after
Movies with an R rating are colored purple, those with a PG-13 rating are red, and those with a PG rating are green. Regression lines have been found for each group. (The black points are G-rated, but there were too few to fit a line reliably.)
Movie Ratings. Does the cost of making a movie depend on its audience? Here s a scatterplot of the same data we examined in Exercise
Movie Dramas. Here s a scatterplot of the production budgets (in millions of dollars) vs. the running time(in minutes) for major release movies in 2005. Dramas are plotted in red and all other genres are plotted in black. A separate least squares regression line has been fitted to each group. For
Bad model? A student who has created a linear model is disappointed to find that her value is a very low 13%.a) Does this mean that a linear model is not appropriate?Explain.b) Does this model allow the student to make accurate predictions? Explain.
Good model? In justifying his choice of a model, a student wrote, I know this is the correct model because .a) Is this reasoning correct? Explain.b) Does this model allow the student to make accurate predictions? Explain.
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