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
122.3	21
194.7	38
167.2	19
82.4	11
134.4	21
157.1	29
173.3	54
81.4	12
147.2	27
112.3	11
84.5	13
164.4	40
204.4	49
173.8	32
157.9	23
206.1	40
222.5	44
222.5	57
232.5	38
190.3	37
99.6	8

A. 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.

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

C. What conclusions can be reached concerning the relationship of the diameter of the tree and its​ height?

Answer 1

Using Excel, go to Data, select Data Analysis, choose Regression. Put height in Y input range and diameter in X input range.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.865
R Square	0.748
Adjusted R Square	0.734
Standard Error	24.649
Observations	21
ANOVA
	df	SS	MS	F	Significance F
Regression	1	34218.634	34218.634	56.322	0.000
Residual	19	11543.524	607.554
Total	20	45762.158
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	75.766	12.279	6.170	0.000	50.065	101.467
Diameter	2.788	0.371	7.505	0.000	2.010	3.566

A. H₀: β₁ = 0, There is no significant relationship between the height of trees of this species and the​breast-height diameter

H₁: β₁ ≠ 0, There is a significant relationship between the height of trees of this species and the​breast-height diameter

Test statistic (t Stat for Diameter) = 7.505

p-value (Hours of training) = 0.000

Level of significance = 0.05

Since p-value is less than 0.05, we reject the null hypothesis and conclude that β₁ ≠ 0.

So, there is a significant relationship between the height of trees of this species and the​breast-height diameter.

B. 95​% confidence interval estimate of the population slope between the height of the trees and​ breast-height diameter: (2.010, 3.566)

C. There is a significant relationship between the height of trees of this species and the​ breast-height diameter.

Since slope = 2.788, one unit increase in breast-height diameter increases height of tree by 2.788 units.