Question

Measuring the height of a California redwood tree is very difficult because these trees grow to heights of 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. The data in Redwood represent the height ( in feet) and diameter ( in inches) at the breast height of a person for a sample of 21 California redwood trees.

a. Assuming a linear relationship, use the least- squares method to compute the regression coefficients b0 and b1. State the regression equation that predicts the height of a tree based on the tree’s diameter at breast height of a person.

b. Interpret the meaning of the slope in this equation.

c. Predict the mean height for a tree that has a breast height diameter of 25 inches.

d. Interpret the meaning of the coefficient of determination in this problem.

e. Perform a residual analysis of the results and determine the adequacy of the model.

f. Determine whether there is a significant relationship between the height of redwood trees and the breast height diameter at the 0.05 level of significance.

g. Construct a 95% confidence interval estimate of the population slope between the height of the redwood trees and breast height diameter.

h. What conclusion can you reach about the relationship between the diameter of the tree and its height?

Height	Diameter at breast height	Bark thickness
122.0	20	1.1
193.5	36	2.8
166.5	18	2.0
82.0	10	1.2
133.5	21	2.0
156.0	29	1.4
172.5	51	1.8
81.0	11	1.1
148.0	26	2.5
113.0	12	1.5
84.0	13	1.4
164.0	40	2.3
203.3	52	2.0
174.0	30	2.5
159.0	22	3.0
205.0	42	2.6
223.5	45	4.3
195.0	54	4.0
232.5	39	2.2
190.5	36	3.5
100.0	8	1.4

Answer 1

After performing regression in MINITAB, results are given below:

a). the coefficient are b0 = 78.8, b1=2.673.

b). unit increase in diameter will cause 2.673 units increases in height.

c).height = 78.8 +2.673 (25) = 145.625

d).71.46% of variation in height is explained by the model.

f).since the p-value is 0.000 < 0.05. the model is significant.

g). the confidence interval is given in the above output.

h) there is a significant relationship between height and diameter.

e). residual plots are given below: