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. 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 35 inches.

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

E. Perform a residual analysis on the results and determine the adequacy of the fit of the model.

F. 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.

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

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

Answer 1

Using Excel<data<data analysis<regression

Regression Analysis
	r²	0.748
	r	0.865
	Std. Error	24.649
	n	21
	k	1
	Dep. Var.	Height
ANOVA table
Source	SS	df	MS	F	p-value
Regression	34,218.6343	1	34,218.6343	56.32	4.27E-07
Residual	11,543.5238	19	607.5539
Total	45,762.1581	20
Regression output				confidence interval
variables	coefficients	std. error	t (df=19)	p-value	95% lower	95% upper
Intercept	75.7662	12.28	6.17	0.00	50.07	101.47
Diameter at breast height	2.7880	0.3715	7.505	4.27E-07	2.0104	3.5655
Predicted values for: Height
		95% Confidence Interval	95% Prediction Interval
Diameter at breast height	Predicted	lower	upper	lower	upper	Leverage
35	173.3461	161.3614	185.3307	120.3822	226.3100	0.054

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=75.7662

b1=2.7880

The regression equation is:

Height=757662+2.7880* Diameter at breast height

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

The 1 unit increase in height will increase the diameter at breast height by 2.2880 units.

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

The regression equation is:

Height=757662+2.7880* Diameter at breast height

Put breast-height diameter=35

Height=757662+2.7880* 35=173.3461 inches

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

The coefficient of determnination=0.748

The coefficient of determination tells us that due to Diameter at breast height 74.8% variation in Height.

E. Perform a residual analysis on the results and determine the adequacy of the fit of the model.

Observation	Height	Predicted	Residual
1	122.30	134.31	-12.01
2	194.70	181.71	12.99
3	167.20	128.74	38.46
4	82.40	106.43	-24.03
5	134.40	134.31	0.09
6	157.10	156.62	0.48
7	173.30	226.32	-53.02
8	81.40	109.22	-27.82
9	147.20	151.04	-3.84
10	112.30	106.43	5.87
11	84.50	112.01	-27.51
12	164.40	187.29	-22.89
13	204.40	212.38	-7.98
14	173.80	164.98	8.82
15	157.90	139.89	18.01
16	206.10	187.29	18.81
17	222.50	198.44	24.06
18	222.50	234.68	-12.18
19	232.50	181.71	50.79
20	190.30	178.92	11.38
21	99.60	98.07	1.53

F. 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.

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

Since p-value(0.000)<alpha(0.05).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.

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

Since r=0.865. This implies that there is a strong positive relationship between the diameter of the tree and its​ height as diameter of tree increases then height increases and if the diameter of tree decreases then height decreases.