Measuring the height of a California redwood tree is very difficult because these trees grow to heights over 300 feet. People familiar with these trees understand that the height of a California redwood tree is related to other characteristics of the tree, including the diameter of the tree at the breast height of a person (in inches) and the thickness of the bark of the tree (in inches). The file Redwood contains the height, diameter at breast height of a person, and bark thickness for a sample of 21 California redwood trees.
a. State the multiple regression equation that predicts the height of a tree, based on the tree's diameter at breast height and the thickness of the bark.
b. Interpret the meaning of the slopes in this equation.
c. Predict the height for a tree that has a breast height diameter of 25 inches and a bark thickness of 2 inches.
d. Interpret the meaning of the coefficient of multiple determinations in this problem.
e. Perform a residual analysis on the results and determine whether the regression assumptions are valid.
f. Determine whether there is a significant relationship between the height of redwood trees and the two independent variables (breast-height diameter and bark thickness) at the 0.05 level of significance.
g. Construct a 95% confidence interval estimate of the population slope between the height of redwood trees and breast-height diameter and between the height of redwood trees and the bark thickness.
h. At the 0.05 level of significance, determine whether each independent variable makes a significant contribution to the regression model. Indicate the independent variables to include in this model.
i. Construct a 95% confidence interval estimate of the mean height for trees that have a breast-height diameter of 25 inches and a bark thickness of 2 inches, along with a prediction interval for an individual tree.
j. Compute and interpret the coefficients of partial determination.