A sample of n = 61 penguin burrows was selected, and values of both y = trail length (m) and x 5 soil hardness (force required to penetrate the substrate to a depth of 12 cm with a certain gauge, in kg) were determined for each one (“Effects of Substrate on the Distribution of Magellanic Penguin Burrows,” The Auk : 923– 933). The equation of the least-squares line was y^ = 11.607 2 1.4187x, and r2 = .386.
a. Does the relationship between soil hardness and trail length appear to be linear, with shorter trails associated with harder soil (as the article asserted)? Carry out an appropriate test of hypotheses.
b. Using se = 2.35, x = 4.5, and g 1x 2 x2 2 = 250, predict trail length when soil hardness is 6.0 in a way that conveys information about the reliability and precision of the prediction.
c. Would you use the simple linear regression model to predict trail length when hardness is 10.0? Explain your reasoning.