Question

Measuring the height of a particular species of tree is very difficult because these trees grow to tremendous heights. People familiar with these trees understand that the height of a tree of this species is related to other characteristics of the​ tree, including the diameter of the tree at the breast height of a person. The accompanying data represent the height​ (in feet) and diameter​ (in inches) at the breast height of a person for a sample of 21 trees of this species.

Height   Diameter at breast height
121.6   21
194.7   38
167.1   19
81.7   11
133.2   20
155.9   28
172.8   54
80.4   10
147.9   26
112.4   12
84.1   12
163.6   40
202.4   55
174.6   32
158.7   23
206.5   44
223.8   47
193.4   54
230.8   41
189.5   36
100.1   8

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. Predict the mean height for a tree that has a​ breast-height diameter of 35 inches.

C. Interpret the meaning of the coefficient of determination in this problem. The value is?

D. Determine whether there is a significant relationship between the height of trees of this species and the​breast-height diameter at the 0.05 level of significance.

- Identify the t Stat value for Diameter at Breast​ Height, rounding to two decimal places.

E. Construct a 95​% confidence interval estimate of the population slope between the height of the trees and​breast-height diameter.

Answer 1

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.

b0 = 79.9345

b1 = 2.5619

The regression equation is:

y = 79.9345 + 2.5619*x

B. Predict the mean height for a tree that has a​ breast-height diameter of 35 inches.

y = 79.9345 + 2.5619*35 = 169.60

C. Interpret the meaning of the coefficient of determination in this problem. The value is?

r2 = 0.742

74.2% of the variation in the model is explained.

D. Determine whether there is a significant relationship between the height of trees of this species and the​breast-height diameter at the 0.05 level of significance.

- Identify the t Stat value for Diameter at Breast​ Height, rounding to two decimal places.

The hypothesis being tested is:

H₀: β₁ = 0

H₁: β₁ ≠ 0

t = 7.39

The p-value is 0.000.

Since the p-value (0.000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that the relationship is significant.

E. Construct a 95​% confidence interval estimate of the population slope between the height of the trees and​breast-height diameter.

The confidence interval estimate of the population slope between the height of the trees and​breast-height diameter is between 1.8364 and 3.2874.

	r²	0.742
	r	0.861
	Std. Error	24.142
	n	21
	k	1
	Dep. Var.	Height
ANOVA table
Source	SS	df	MS	F	p-value
Regression	31,839.0793	1	31,839.0793	54.63	5.32E-07
Residual	11,073.6664	19	582.8245
Total	42,912.7457	20
Regression output				confidence interval
variables	coefficients	std. error	t (df=19)	p-value	95% lower	95% upper
Intercept	79.9345
Diameter	2.5619	0.3466	7.391	5.32E-07	1.8364	3.2874