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 = sunshine index. The cloud cover index can have values between 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.

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.

ŷ = 10.8768 + 1.4604x - 7.259x² + 9.2342x³ 

c. Construct a residual plot by plotting the residuals from the cubic regression model versus x. Are there any troubling patterns in the residual plot that suggest that a cubic regression is not an appropriate way to summarize the relationship?

d. Use the cubic regression to predict sunshine index when the cloud cover index is 0.25.

e. Use the cubic regression to predict sunshine index when the cloud cover index is 0.45.

f. Explain why it would not be a good idea to use the cubic regression equation to predict sunshine index for a cloud cover index of 0.75.

Transcribed Image Text:

Cloud Cover Index (x) 0.2 0.5 0.3 0.1 0.2 0.4 0.0 0.4 0.3 Sunshine Index (y) 10.98 10.94 10.91 10.94 10.97 10.89 10.88 10.92 10.86