Chapter 14, Problem 75PS

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 hight of a person, and bank 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 thickness of the bark.

b. Interpret the meaning of the slopes in the equation.

c. Predict the mean 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 determination in this problem.

e. Perform a residual analysis on the model 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 variables 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 the interpret the coefficients of partial determination.

k. What conclusions can you reach concerning the effect of the diameter of tree and the thickness of the bark on the height of the tree?