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
122.5   19
193.7   38
166.4   18
82.7   11
132.9   21
156.1   28
171.8   49
80.9   11
147.6   25
113.3   12
84.5   12
165.2   42
204.2   54
172.7   29
158.9   23
206.4   42
222.4   47
195.1   51
234.1   41
189.7   38
99.9   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

Let X = Height

Y = Diameter of Breast Height

The Regression Equation of Y on X is

Where

  is the Regression Coefficient of Y on X

The Regression Equation of X on Y is  

Where is the Regression Coefficient of X on Y

	X	Y
	122.5	19	-34.6905	-10.4762	1203.4291	109.7506	363.42404
	193.7	38	36.5095	8.5238	1332.9453	72.65533	311.20023
	166.4	18	9.2095	-11.4762	84.815329	131.7029	-105.69025
	82.7	11	-74.4905	-18.4762	5548.831	341.3696	1376.3002
	132.9	21	-24.2905	-8.4762	590.02723	71.8458	205.8907
	156.1	28	-1.0905	-1.4762	1.1891383	2.179138	1.6097506
	171.8	49	14.6095	19.5238	213.43819	381.1791	285.23356
	80.9	11	-76.2905	-18.4762	5820.2368	341.3696	1409.5574
	147.6	25	-9.5905	-4.4762	91.977234	20.03628	42.928798
	113.3	12	-43.8905	-17.4762	1926.3739	305.4172	767.03832
	84.5	12	-72.6905	-17.4762	5283.9053	305.4172	1270.3526
	165.2	42	8.0095	12.5238	64.152472	156.8458	100.30975
	204.2	54	47.0095	24.5238	2209.8953	601.4172	1152.8526
	172.7	29	15.5095	-0.4762	240.54533	0.226757	-7.3854875
	158.9	23	1.7095	-6.4762	2.9224717	41.94104	-11.071202
	206.4	42	49.2095	12.5238	2421.5772	156.8458	616.2907
	222.4	47	65.2095	17.5238	4252.282	307.0839	1142.7193
	195.1	51	37.9095	21.5238	1437.132	463.2744	815.95737
	234.1	41	76.9095	11.5238	5915.0749	132.7982	886.2907
	189.7	38	32.5095	8.5238	1056.8691	72.65533	277.10499
	99.9	8	-57.2905	-21.4762	3282.1987	461.2268	1230.3812
	157.1905	29.4762	0.0000	0.0000	42979.8181	4477.2381	12131.2952

(A) The Regression Equations:

Therefore Regression Equation of Y on X is

The Regression Equation of X on Y

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(B) Given Y = 35; X = ?

We use the Regression Equaion of X on Y

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(C) COEFFICIENT OF DETERMINATION:

Therefore 76.48% variation of dependent variable Y is being expressed by the independent variable X. More the Coefficient of determination; good linear relation exists between the two variables.

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(D) We fram the Null Hypothesis

: There is no significant relationship between the height of trees of this species and the​breast-height diameter

To test the we use the t - Statistic

Therefore

Since = 0.05 = 5%

Since ; So, We Reject a5 5% Level of Significant

Therefore we Conclude that "There is significant relationship between the height of trees of this species and the​breast-height diameter".

NOTE: ;The citical value of t has been extracted from the tabulated vallueof t which is posted below.