5.33 Toucan’s beak.Exercise 4.45 (page 125) gives data on beak heat loss, as a percent of total body heat loss from all sources, at various temperatures. The data show that beak heat loss is higher at higher temperatures and that the relationship is roughly linear. Figure 5.12 shows Minitab regression output for these data.
Figure 5.12Minitab regression output for a study of how temperature affects beak heat loss in toucans, for Exercise 5.33.
(a)What is the equation of the least-squares regression line for predicting beak heat loss, as a percent of total body heat loss from all sources, from temperature? Explain in specific language what this slope says about the relationship between beak heat loss and temperature.
(b)Use the equation of the least-squares regression line to predict beak heat loss, as a percent of total body heat loss from all sources, at a temperature of 25 degrees Celsius.
(c)What percent of the variation in beak heat loss is explained by the straight-line relationship with temperature?
(d)Use the information in Figure 5.12 to find the correlation r between beak heat loss and temperature. How do you know whether the sign of r is positive or negative?
5.41 Our brains don’t like losses.Exercise 4.29 (page 121) describes an experiment that showed a linear relationship between how sensitive people are to monetary losses (“behavioral loss aversion”) and activity in one part of their brains (“neural loss aversion”).
(b)Find the least-squares line for predicting y from x, leaving out the outlier, and add the line to your plot.
(c)The outlier lies very close to your regression line. Looking at the plot, you now expect that adding the outlier will increase the correlation but will have little effect on the least-squares line. Explain why.
(d)Find the correlation with and without the outlier. Your results verify the expectations from part (c).
5.42 Always plot your data!Table 5.1 presents four sets of data prepared by the statistician Frank Anscombe to illustrate the dangers of calculating without first plotting the data.19
(a)Without making scatterplots, find the correlation and the least-squares regression line for all four data sets. What do you notice? Use the regression line to predict y for x = 10.
(c)In which of the four cases would you be willing to use the regression line to describe the dependence of y on x? Explain your answer in each case.
The following exercises ask you to answer questions from data without having the details outlined for you. The exercise statements give you theStatestep of the four-step process. In your work, follow thePlan, Solve,andConcludesteps of the process, described on page 63.
5.61 Predicting tropical storms. William Gray heads the Tropical Meteorology Project at Colorado State University (well away from the hurricane belt). His forecasts before each year’s hurricane season attract lots of attention. Here are data on the number of named Atlantic tropical storms predicted by Dr. Gray and the actual number of storms for the years 1984 to 2013:25
Analyze these data. How accurate are Dr. Gray’s forecasts? How many tropical storms would you expect in a year when his preseason forecast calls for 16 storms? What is the effect of the disastrous 2005 season on your answers?
5.62 Great Arctic rivers. One effect of global warming is to increase the flow of water into the Arctic Ocean from rivers. Such an increase may have major effects on the world’s climate. Six rivers (Yenisey, Lena, Ob, Pechora, Kolyma, and Severnaya Dvina) drain two-thirds of the Arctic in Europe and Asia. Several of these are among the largest rivers on earth. Table 5.4 presents the total discharge from these rivers each year from 1936 to 2010.26 Discharge is measured in cubic kilometers of water. Analyze these data to uncover the nature and strength of the trend in total discharge over time.