Researchers have examined a number of climatic variables in an attempt to understand the mechanisms that govern rainfall runoff. The paper "The Applicability of Morton's and Penman's Evapotranspiration Estimates in Rainfall-Runoff Modeling" (Water Resources Bulletin [1991]: 611-620) reported on a study that examined the relationship between x = cloud cover index and y = sun- shine index. The cloud cover index can have values be- tween 0 and 1. The accompanying data are consistent with summary quantities in the article. The authors of the article used a cubic regression to describe the relationship between cloud cover and sunshine.
Cloud Cover Index (x) Sunshine Index (y)
0.2 10.98
0.5 10.94
0.3 10.91
0.1 10.94
0.2 10.97
0.4 10.89
0.0 10.88
0.4 10.92
0.3 10.86
a. Construct a scatterplot of the data. What characteristics of the plot suggest that a cubic regression would be more appropriate for summarizing the relationship between sunshine index and cloud cover index than a linear or quadratic regression?
b. Find the equation of the least-squares cubic function.
c. Construct a residual plot by plotting the residuals
|
Age
|
Fatality Rate
|
Age
|
Fatality Rate
|
|
40
|
0.75
|
70
|
1.30
|
|
45
|
0.75
|
75
|
1.65
|
|
50
|
0.95
|
80
|
2.20
|
|
55
|
1.05
|
85
|
3.00
|
|
60
|
1.15
|
90
|
3.20
|
|
65
|
1.20
|
|
|
a. Construct a scatterplot of these data.
b. Using Table 5.5 and the ladder of transformations in Figure 5.38, suggest a transformation that might result in variables for which the scatterplot would exhibit a pattern that was more nearly linear.
c. Reexpress x and/or y using the transformation you recommended in Part (b). Construct a scatterplot of the transformed data.
d. Does the scatterplot in Part (c) suggest that the transformation was successful in straightening the plot?
e. Using the transformed variables, ?t the least-squares line and use it to predict the fatality rate for 78-year- old drivers.