Good model In justifying his choice of a model, a student wrote, I know this is the

Each of these scatterplots shows an unusual point. For each, tell whether the point is a high-leverage point, would have a large residual, or is influential. 12y2 4 6 8 10

Each of these scatterplots shows an unusual point. For each, tell whether the point is a high-leverage point, would have a large residual, or is influential. 25 0 1 5 0 15 20

Each of these scatterplots shows an unusual point. For each, tell whether the point is a high-leverage point, would have a large residual, or is influential. 20 15 1 5 0 5 10 15 20 y 0 2 5 10 15 0

HDI 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 phone subscribers per 100 people is positively associated with economic progress in a country. Can the number of phone subscribers be used to predict the HDI? Here is a scatterplot of HDI against phone subscribers: 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?

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.

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.

Movie dramas Heres 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 xs 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?

Smoking 2009, women and men In Exercise 2, we examined the percentage of men aged 1824 who smoked from 1965 to 2009 according to the Centers for Disease Control and Prevention. How about women? Heres 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.)

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 Abalones 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?

Abalone again The researcher in Exercise 9 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?

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?

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?

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 teachers class averages and they should just work with those averages. Explain why this is problematic.

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

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 7.5halfway through the 7th year after 1990.) www.oaklandairport.com Look at the regression below and the residuals plot on the next page. a) Interpret the slope and intercept of the model. b) What does the value of R2 say about the model? c) Interpret se in this context. d) Would you use this model to predict the numbers of passengers in 2010 (YearsSince1990 = 20) Explain. e) Theres a point near the middle of this time span with a large negative residual. Can you explain this outlier?

Tracking hurricanes 2010 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 below shows the trend in the 24-hour tracking errors since 1970 (www.nhc.noaa.gov). T 0 10 20 30 40 Years Since 1970 125 100 75 50 Error Dependent variable is Error R@squared = 68.7, s = 16.44 Variable Coefficient Intercept 132.301 Years - 1970 -2.00662 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?

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.

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.

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 (ae) with the correct new correlation from the list given. 1) -0.90 2) -0.40 3) 0.00

The extra point revisited The original five points in Exercise 19 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) -0.45 2) -0.30 3) 0.00 4) 0.05 5) 0.85

Whats 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.

Whats the effect? A researcher studying violent behavior in elementary school children asks the childrens 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.

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 schools 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 dont bother.) What value of the slope of that line would you view as demonstrating acceptable progress in reading comprehension? Explain.

Grades A college admissions officer, defending the colleges use of SAT scores in the admissions process, produced the graph below. It shows the mean GPAs for last years 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?

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 Cost = 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.

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: a) Interpret the slope of this line in context. b) Explain why its 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.

Interest rates Heres 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.

Marriage Age 2010 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? Heres a timeplot showing the difference in average age (mens age - womens 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.

Interest rates revisited In Exercise 27 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. Heres a regression model for the data since 1980. Dependent variable is: Rate R-squared = 74.5% s = 1.630 Variable Coefficient Intercept 21.0688 Year 1950 -0.356578 a) How does this model compare to the one in Exercise 27? b) What does this model estimate the interest rate to have been in 2000? How does this compare to the rate you predicted in Exercise 27? c) Do you trust this newer predicted value? Explain. d) Given these two models, what would you predict the interest rate on 3-month Treasury bills will be in 2020?

Ages of couples again Has the trend of decreasing difference in age at first marriage seen in Exercise 28 gotten stronger recently? The scatterplot and residual plot for the data from 1980 through 2010, 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 its not reasonable to interpret the y-intercept.

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 animals 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. 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.

Swim the lake 2010 People swam across Lake Ontario 48 times between 1974 and 2010 (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 plot: T Dependent variable is Time R@squared = 2.0, s = 449.9 Variable Coefficient Intercept -9943.083 Year 5.64227 1,500 750 1200 1250 1300 Predicted Residuals 1350 20 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 doesnt belong with the others. Would removing it change the model a lot? Explain.

Elephants and hippos We removed humans from the scatterplot in Exercise 31 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.

Another swim 2010 In Exercise 32, we saw that Vicki Keiths 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: T Dependent variable is Time R@Squared = 6.0, s = 326.1 Variable Coefficient Intercept -13123.2 Year 7.21677 a) In this new model, the value of se 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)?

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 1 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.

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.

Life expectancy 2010 Data for 24 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 these two variables and describe the association. b) Find the equation of the regression line. c) Interpret the value of R2 . d) Make a plot of the residuals. Are any countries unusual? e) Is the line an appropriate model? f) If you conclude that there is an outlier, set it aside and recompute the regression. g) If government leaders want to increase life expectancy, should they encourage women to have fewer children? Explain.

Tour de France 2011 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). In 2005, Lance Armstrong averaged 41.65 kph (25.88 mph) for the fastest average speed in history. (Later, of course, his record was vacated when he admitted doping.) 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.

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. 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.

Second stage 2011 Look once more at the data from the Tour de France. In Exercise 38, we looked at the whole history of the race, but now lets consider just the postWorld War II era. a) Find the regression of Avg Speed by Year only for years from 1947 to the present. Are the conditions for regression met? b) Interpret the slope. c) In 1979, Bernard Hinault averaged 39.8 kph, while in 2005 Lance Armstrong averaged 41.65 kph. Which was the more remarkable performance and why?

Bridges covered There is a relationship between the age of a bridge in Tompkins County, New York, and its condition as found by inspection. Below is a graph of data. Two of bridges are considerably older than the rest. The oldest covered bridge in daily use in New York State was built in 1853 and was recently judged to have a condition of 4.523. Another bridge with a rating of 5.4 was built in 1865. Also below is the regression analysis for this relationship using all 196 bridges. 1850 1900 1950 2000 Year built 3 4 5 6 7 HDI Dependent variable is Condition R@squared = 42.54, s = 0.7603 Variable Coefficient Constant -36.0501 Year 0.021048 a) If we use this regression to predict the condition of the oldest bridge, what would its residual be? We removed the 1853 and 1865 bridges from the analysis and here are the results: Dependent variable is Condition R@squared = 46.55, s = 0.7354 Variable Coefficient Constant -40.7546 Year 0.02343 b) How were these two old bridges affecting the regression? c) The 1853 bridge was extensively restored in 1972. Does that better explain its condition?