Credit card spending An analysis of spending by a sample

Problem 11E Skinned knees There is a strong correlation between the temperature and the number of skinned knees on playgrounds. Does this tell us that warm weather causes children to trip?

Problem 13E Grading A team of Calculus teachers is analyzing student scores on a final exam compared to the midterm scores. One teacher proposes that they already have every teacher’s class averages and they should just work with those averages. Explain why this is problematic.

Problem 8E Revenue and advanced sales The production company of Exercise 7 offers advanced sales to “Frequent Buyers” through its website. Here’s a relevant scatterplot: Exercise 7: Revenue and large venues A regression of Total Revenue on Ticket Sales by the concert production company of Exercises 2 and 4 finds the model a) Management is considering adding a stadium-style venue that would seat 10,000. What does this model predict that revenue would be if the new venue were to sell out? b) Why would it be unwise to assume that this model accurately predicts revenue for this situation? One performer refused to permit advanced sales. What effect has that point had on the regression to model Total Revenue from Advanced Sales?

Problem 12E Cell phones and life expectancy The correlation between cell phone usage and life expectancy is very high. Should we buy cell phones to help people live longer?

Problem 10E Abalone again The researcher in Exercise is content with the second regression. But he has found a number of shells that have large residuals and is considering removing all of them. Is this good practice? Exercise Abalone Abalones are edible sea snails that include over 100 species. A researcher is working with a model that uses the number of rings in an Abalone’s shell to predict its age. He finds an observation that he believes has been miscalculated. After deleting this outlier, he redoes the calculation. Does it appear that this outlier was exerting very much influence? Before: Dependent variable is Age R@squared = 67.5% Variable Coefficient Intercept 1.736 Rings 0.45 After: Dependent variable is Age R@squared = 83.9% Variable Coefficient Intercept 1.56 Rings 1.13

Problem 9E Abalone Abalones are edible sea snails that include over 100 species. A researcher is working with a model that uses the number of rings in an Abalone’s shell to predict its age. He finds an observation that he believes has been miscalculated. After deleting this outlier, he redoes the calculation. Does it appear that this outlier was exerting very much influence? Before: Dependent variable is Age R@squared = 67.5% Variable Coefficient Intercept 1.736 Rings 0.45 After: Dependent variable is Age R@squared = 83.9% Variable Coefficient Intercept 1.56 Rings 1.13

Problem 14E Average GPA An athletic director proudly states that he has used the average GPAs of the university’s sports teams and is predicting a high graduation rate for the teams. Why is this method unsafe?

Problem 19E Good model? In justifying his choice of a model, a student wrote, “I know this is the correct model because R2 = 99.4%.” a) Is this reasoning correct? Explain. ________________ b) Does this model allow the student to make accurate predictions? Explain.

Smoking 2011 The Centers for Disease Control and Prevention track cigarette smoking in the United States. How has the percentage of people who smoke changed since the danger became clear during the last half of the 20th century? The scatterplot shows percentages of smokers among men 18–24 years of age, as estimated by surveys, from 1965 through 2011 (http://www.cdc .gov/nchs/). a) Is there a clear pattern? Describe the trend. b) Is the association strong? c) Is a linear model appropriate? Explain.

Problem 17E Human Development Index 2012 The United Nations Development Programme (UNDP) uses the Human Development Index (HDI) in an attempt to summarize in one number the progress in health, education, and economics of a country. In 2012, the HDI was as high as 0.955 for Norway and as low as 0.304 for Niger. The gross domestic product per capita (GDPPC), by contrast, is often used to summarize the overall economic strength of a country. Is the HDI related to the GDPPC? At the top of the next column is a scatterplot of HDI against GDPPC. a) Explain why fitting a linear model to these data might be misleading. b) If you fit a linear model to the data, what do you think a scatterplot of residuals versus predicted HDI will look like?

Problem 18E HDI 2012 revisited The United Nations Development Programme (UNDP) uses the Human Development Index (HDI) in an attempt to summarize in one number the progress in health, education, and economics of a country. The number of Internet users per 100 people is positively associated with economic progress in a country. Can the number of Internet users be used to predict the HDI? Here is a scatterplot of HDI against Internet users: a) Explain why fitting a linear model to these data might be misleading. b) If you fit a linear model to the data, what do you think a scatterplot of residuals vs. predicted HDI will look like?

Problem 15E Marriage age 2010 Is there evidence that the age at which women get married has changed over the past 100 years? The scatterplot shows the trend in age at first marriage for American women (www.census.gov). a) Is there a clear pattern? Describe the trend. ________________ b) Is the association strong? ________________ c) Is the correlation high? Explain. ________________ d) Is a linear model appropriate? Explain.

Problem 20E Bad model? A student who has created a linear model is disappointed to find that her R2 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.

Problem 21E 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 as red x’s and all other genres are plotted as blue dots. (The re-make of King Kong is plotted as a black “-”. At the time it was the most expensive movie ever made, and not typical of any genre.) A separate least squares regression line has been fitted to each group. For the following questions, just examine the plot: a) What are the units for the slopes of these lines? ________________ b) In what way are dramas and other movies similar with respect to this relationship? ________________ c) In what way are dramas different from other genres of movies with respect to this relationship?

Problem 22E Smoking 2011, women and men In Exercise 16, we examined the percentage of men aged 18–24 who smoked from 1965 to 2011 according to the Centers for Disease Control and Prevention. How about women? Here’s a scatterplot showing the corresponding percentages for both men and women: a) In what ways are the trends in smoking behavior similar for men and women? b) How do the smoking rates for women differ from those for men? c) Viewed alone, the trend for men may have seemed to violate the Linearity Condition. How about the trend for women? Does the consistency of the two patterns encourage you to think that a linear model for the trend in men might be appropriate? (Note: there is no correct answer to this question; it is raised for you to think about.) Exercise 16: Smoking 2011 The Centers for Disease Control and Prevention track cigarette smoking in the United States. How has the percentage of people who smoke changed since the danger became clear during the last half of the 20th century? The scatterplot shows percentages of smokers among men 18–24 years of age, as estimated by surveys, from 1965 through 2011 (http://www.cdc .gov/nchs/). a) Is there a clear pattern? Describe the trend. b) Is the association strong? c) Is a linear model appropriate? Explain.

Problem 24E Tracking hurricanes 2012 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 scatterplot at the top of the next page shows the trend in the 24-hour tracking errors since 1970 (www.nhc.noaa.gov). a) Interpret the slope and intercept of the model. b) Interpret se in this context. c) The Center would like to achieve an average tracking error of 45 nautical miles by 2015. Will they make it? Defend your response. d) What if their goal were an average tracking error of 25 nautical miles? e) What cautions would you state about your conclusion?

Problem 25E 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 were removed, would the correlation become stronger or weaker? Explain. 4) If that point were removed, would the slope of the regression line increase or decrease? Explain. a) ________________ b) ________________ c) ________________ d)

Problem 26E 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 removed, would the correlation Become stronger or weaker? Explain. 4) If that point were removed, would the slope of the Regression line increase or decrease? Explain. a) ________________ b) ________________ c) ________________ d)

Problem 23E Oakland passengers 2013 The scatterplot below shows the number of passengers at Oakland (CA) airport month by month since 1997. (www.oaklandairport.com) a) Describe the patterns in passengers at Oakland airport that you see in this time plot. b) Until 2009, analysts got fairly good predictions using a linear model. Why might that not be the case now? c) If they considered only the data from 2009 to the present, might they get reasonable predictions into the future?

Problem 28E The extra point revisited The original five points in Exercise produce a regression line with slope 0. Match each of the green points (a–e) with the slope of the line after that one point is added: 1) -0.45 2) -0.30 3) 0.00 4) 0.05 5) 0.85 Exercise The extra point The scatterplot shows five blue data points at the left. Not surprisingly, the correlation for these points is r = 0. Suppose one additional data point is added at one of the five positions suggested below in green. Match each point (a–e) with the correct new correlation from the list given. 1) -0.90 2) -0.40 3) 0.00 4) 0.05 5) 0.75

Problem 29E 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 causeand effect relationships that might be present.

Problem 30E 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 that the researcher finds a moderately strong positive correlation. Describe three different possible cause-and-effect explanations for this relationship.

Problem 27E The extra point The scatterplot shows five blue data points at the left. Not surprisingly, the correlation for these points is r = 0. Suppose one additional data point is added at one of the five positions suggested below in green. Match each point (a–e) with the correct new correlation from the list given. 1) -0.90 2) -0.40 3) 0.00 4) 0.05 5) 0.75

Problem 33E 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 with the model = 133 - 2.13 Temp. Here is the residuals plot for his data: a) Interpret the slope of the line in this context. ________________ b) Interpret the y-intercept of the line in this context. ________________ c) During months when the temperature stays around freezing, would you expect cost predictions based on this model to be accurate, too low, or too high? Explain. ________________ d) What heating cost does the model predict for a month that averages 10°? ________________ e) During one of the months on which the model was based, the temperature did average 10°. What were the actual heating costs for that month? ________________ f) Should the homeowner use this model? Explain. ________________ g) Would this model be more successful if the temperature were expressed in degrees Celsius? Explain.

Problem 31E 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 principal prepares a report to parents that includes a graph showing the mean reading score for each grade. In his comments he points out that the strong positive trend demonstrates the success of the school’s reading program. a) Does this graph indicate that students are making satisfactory progress in reading? Explain. ________________ b) What would you estimate the correlation between Grade and Average Reading Level to be? ________________ c) If, instead of this plot showing average reading levels, the principal had produced a scatterplot of the reading levels of all the individual students, would you expect the correlation to be the same, higher, or lower? Explain. ________________ d) Although the principal did not do a regression analysis, someone as statistically astute as you might do that. (But don’t bother.) What value of the slope of that line would you view as demonstrating acceptable progress in reading comprehension? Explain.

Problem 34E 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 Speed and created this Residual plot: a) Interpret the slope of this line in context. ________________ b) Explain why it’s silly to attach any meaning to the y-intercept. ________________ c) When this model predicts high Fuel Efficiency, what can you say about those predictions? ________________ d) What Fuel Efficiency does the model predict when the car is driven at 50 mph? ________________ e) What was the actual Fuel Efficiency when the car was driven at 45 mph? ________________ f) Do you think there appears to be a strong association between Speed and Fuel Efficiency? Explain. ________________ g) Do you think this is the appropriate model for that association? Explain.

Problem 32E 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 the graph, the statistical methodology or the conclusions reached?

Interest rates 2014 Here’s a plot showing the federal rate on 3-month Treasury bills from 1950 to 1980, and a regression model fit to the relationship between the Rate (in %) and Years Since 1950 (www.gpoaccess .gov/eop/). a) What is the correlation between Rate and Year? b) Interpret the slope and intercept. c) What does this model predict for the interest rate in the year 2000? d) Would you expect this prediction to have been accurate? Explain.

Problem 37E Interest rates 2014 revisited In Exercise 35, 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 computed a new regression model for the years 1981 to 2013. a) How does this model compare to the one in Exercise 35? b) What does this model estimate the interest rate was in 2000? How does this compare to the rate you predicted in Exercise 35? c) Do you trust this newer predicted value? Explain. d) What does each of the two models predict the interest rate on 3-month Treasury bills will be in 2020? e) Comment on your predictions in d) Exercise 35: Interest rates 2014 Here’s a plot showing the federal rate on 3-month Treasury bills from 1950 to 1980, and a regression model fit to the relationship between the Rate (in %) and Years Since 1950 (www.gpoaccess .gov/eop/). a) What is the correlation between Rate and Year? b) Interpret the slope and intercept. c) What does this model predict for the interest rate in the year 2000? d) Would you expect this prediction to have been accurate? Explain.

Problem 36E Marriage age, 2011 The graph shows the ages of both men and women at first marriage (www.census.gov). Clearly, the patterns for men and women are similar. But are the two lines getting closer together? Here’s a timeplot showing the difference in average age (men’s age - women’s age) at first marriage, the regression analysis, and the associated residuals plot. a) What is the correlation between Age Difference and Year? b) Interpret the slope of this line. c) Predict the average age difference in 2015. d) Describe reasons why you might not place much faith in that prediction.

Marriage age 2011 again Has the trend of decreasing difference in age at first marriage seen in Exercise 36 gotten stronger recently? The scatterplot and residual plot for the data from 1980 through 2011, along with a regression for just those years, are below. a) Is this linear model appropriate for the post-1980 data? Explain. b) What does the slope say about marriage ages? c) Explain why it’s not reasonable to interpret the y-intercept.

Problem 40E Swim the lake 2013 People swam across Lake Ontario 52 times between 1974 and 2013 (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 since 1974 of the crossing and the residuals plot: a) What does the R2 mean for this regression? b) Are the swimmers getting faster or slower? Explain. c) The outlier seen in the residuals plot is a crossing by Vicki Keith in 1987 in which she swam a round trip, north to south, and then back again. Clearly, this swim doesn’t belong with the others. Would removing it change the model a lot? Explain.

Problem 41E Elephants and hippos We removed humans from the scatterplot in Exercise 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 the far right is for hippos. a) By removing one of these points, we could make the association appear to be stronger. Which point? Explain. ________________ b) Would the slope of the line increase or decrease? ________________ c) Should we just keep removing animals to increase the strength of the model? Explain. ________________ d) If we remove elephants from the scatterplot, the slope of the regression line becomes 11.6 days per year. Do you think elephants were an influential point? Explain. Exercise 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 years) for 18 species of mammals. The highlighted point at the far right represents humans. a) For these data, r = 0.54, not a very strong relationship. Do you think the association would be stronger or weaker if humans were removed? Explain. ________________ b) Is there reasonable justification for removing humans from the data set? Explain. ________________ c) Here are the scatterplot and regression analysis for the 17 nonhuman species. Comment on the strength of the association. Dependent variable is: Gestation R@Squared = 72.2% Variable Coefficient Constant -39.5172 LifExp 15.4980 ________________ d) Interpret the slope of the line. ________________ e) Some species of monkeys have a life expectancy of about 20 years. Estimate the expected gestation period of one of these monkeys.

Another swim 2013 In Exercise 40, 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: a) In this new model, the value of \(s_e\) is smaller. Explain what that means in this context. b) Now would you be willing to say that the Lake Ontario swimmers are getting faster (or slower)?

Problem 39E 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 years) for 18 species of mammals. The highlighted point at the far right represents humans. a) For these data, r = 0.54, not a very strong relationship. Do you think the association would be stronger or weaker if humans were removed? Explain. ________________ b) Is there reasonable justification for removing humans from the data set? Explain. ________________ c) Here are the scatterplot and regression analysis for the 17 nonhuman species. Comment on the strength of the association. Dependent variable is: Gestation R@Squared = 72.2% Variable Coefficient Constant -39.5172 LifExp 15.4980 ________________ d) Interpret the slope of the line. ________________ e) Some species of monkeys have a life expectancy of about 20 years. Estimate the expected gestation period of one of these monkeys.

Marriage age 2010 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 15 to get a good prediction? Marriage ages in selected years starting in 1900 are listed below. Use all or part of these data to create an appropriate model for predicting the average age at which women will first marry in 2020. b) How much faith do you place in this prediction? Explain. c) Do you think your model would produce an accurate prediction about your grandchildren, say, 50 years from now? Explain.

Problem 46E Tour de France 2014 We met the Tour de France data set in Chapter 1 (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). By the 21st century, riders were averaging over 40 kph (nearly 25 mph). a) Make a scatterplot of Avg Speed against Year. Describe the relationship of Avg Speed by Year, being careful to point out any unusual features in the plot. b) Find the regression equation of Avg Speed on Year. c) Are the conditions for regression met? Comment.

Bridges covered In Chapter 7, (Data in Tompkins County Bridges 2014) we found a relationship between the age of a bridge in Tompkins County, New York, and its condition as found by inspection. But we considered only bridges built or replaced since 1886. Tompkins County is the home of the oldest covered bridge in daily use in New York State. Built in 1853, it was recently judged to have a condition of 4.57. a) If we use this regression to predict the condition of the covered bridge, what would its residual be? b) If we add the covered bridge to the data, what would you expect to happen to the regression slope? Explain. c) If we add the covered bridge to the data, what would you expect to happen to the \(R^2\) ? Explain. d) The bridge was extensively restored in 1972. If we use that date instead, do you find the condition of the bridge remarkable?

Problem 47E Inflation 2011 The Consumer Price Index (CPI) tracks the prices of consumer goods in the United States, as shown in the following table. The CPI is reported monthly, but we can look at selected values. The table shows the January CPI at five-year intervals. It indicates, for example, that the average item costing $17.90 in 1926 cost $220.22 in the year 2011. Year JanCPI Year JanCPI 1916 10.4 1966 31.8 1921 19.0 1971 39.8 1926 17.9 1976 55.6 1931 15.9 1981 87.0 1936 13.8 1986 109.6 1941 14.1 1991 134.6 1946 18.2 1996 154.4 1951 25.4 2001 175.1 1956 26.8 2006 198.3 1961 29.8 2011 220.223 a) Make a scatterplot showing the trend in consumer prices. Describe what you see. ________________ b) Be an economic forecaster: Project increases in the cost of living over the next decade. Justify decisions you make in creating your model.

Problem 45E Life expectancy 2013 Data for 26 Western Hemisphere countries can be used to examine the association between life expectancy and the birth rate (number of births per 1000 population). a) Create a scatterplot relating Life Expectancy to Birth Rate and describe the association. Are the regression assumptions satisfied? b) Is there an outlier? If so, set it aside and identify it before computing the regression. c) Find the equation of the regression line with the outlier removed. d) Make a plot of the residuals and comment. e) Is the line an appropriate model?. f) Interpret the value of R2. g) If government leaders want to increase life expectancy, should they encourage women to have fewer children? Explain.

Problem 1E Credit card spending An analysis of spending by a sample of credit card bank cardholders shows that spending by cardholders in January (Jan) is related to their spending in December (Dec): The assumptions and conditions of the linear regression seemed to be satisfied and an analyst was about to predict January spending using the model Another analyst worried that different types of cardholders might behave differently. She examined the spending patterns of the cardholders and placed them into five market Segments. When she plotted the data using different colors and symbols for the five different segments, she found the following: Look at this plot carefully and discuss why she might be worried about the predictions from the model

Problem 2E Revenue and talent cost A concert production company examined its records. The manager made the following scatterplot. The company places concerts in two venues, a smaller, more intimate theater (plotted with blue circles) and a larger auditorium-style venue (red x’s). a) Describe the relationship between Talent Cost and Total Revenue. (Remember: direction, form, strength, outliers.) b) How are the results for the two venues similar? c) How are they different?

Revenue and ticket sales The concert production company of Exercise 2 made a second scatterplot, this time relating Total Revenue to Ticket Sales. a) Describe the relationship between Ticket Sales and Total Revenue. b) How are the results for the two venues similar? c) How are they different?

Cell phone costs Noting a recent study predicting the increase in cell phone costs, a friend remarks that by the time he’s a grandfather, no one will be able to afford a cell phone. Explain where his thinking went awry.

Problem 6E Stopping times Using data from 20 compact cars, a consumer group develops a model that predicts the stopping time for a vehicle by using its weight. You consider using this model to predict the stopping time for your large SUV. Explain why this is not advisable.

Revenue and advanced sales The production company of Exercise 7 offers advanced sales to “Frequent Buyers” through its website. Here’s a relevant scatterplot: One performer refused to permit advanced sales. What effect has that point had on the regression to model Total Revenue from Advanced Sales?

Revenue and large venues A regression of Total Revenue on Ticket Sales by the concert production company of Exercises 2 and 4 finds the model \(\widehat {Revenue} = -14,228 + 36.87\) Ticket Sales a) Management is considering adding a stadium-style venue that would seat 10,000. What does this model predict that revenue would be if the new venue were to sell out? b) Why would it be unwise to assume that this model accurately predicts revenue for this situation?