In: Statistics and Probability
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 thebreast-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 andbreast-height diameter.
C. What conclusions can be reached concerning the relationship of the diameter of the tree and its height?
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 thebreast-height diameter
H1: β1 ≠ 0, There is a significant relationship between the height of trees of this species and thebreast-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 thebreast-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.