Question

2. Measuring the height of a California redwood tree is very difficult because these trees grow to heights of over 300 feet. People familiar with these trees understand that the height of the tree is related to other characteristics of the tree, such as the diameter of the tree at the breast height of a person. In the Excel data file for this assignment, under the tab Redwood, is a random sample of 21 redwood trees including the height (in feet) and the diameter (in inches) at breast height.

a. Create the appropriate scatter plot and calculate the coefficient of correlation. Comment on the results. Does it appear a linear relationship exists?

b. Use the least-squares method to compute the regression coefficients b₀ and b₁.

c. Interpret the meaning of b₀ and b₁ in this problem.

d. What if the predicted mean height for a tree that has a breast height diameter of 25 inches? 60 inches?  

e. Determine the coefficient of determination, R², and explain its meaning in this problem.  

g. Perform a complete residual analysis. Does the analysis support all the assumptions required for a valid model? Explain.

h. Create the 95% confidence interval and the 95% prediction interval for a tree with a diameter of 25”. Interpret your results.

i. What conclusions can you reach about the relationship between the diameter of the tree and its height?

Height	Diameter at breast height
122.0	20
193.5	36
166.5	18
82.0	10
133.5	21
156.0	29
172.5	51
81.0	11
148.0	26
113.0	12
84.0	13
164.0	40
203.3	52
174.0	30
159.0	22
205.0	42
223.5	45
195.0	54
232.5	39
190.5	36
100.0	8

Answer 1

Answer:

To solve the problem excel is used

The diameter is entered in column A and the height in column B.

Scatter plot is

The correlation is calculated by using the "CORREL" function in excel.

correlation coefficient=CORREL(A2:A22,B2:B22) = 0.853734

Thus there exists a strong positive linear relationship between the variables under study.

From the scatter plot there appears a strong linear relationship exists.

b) Let us denote diameter by x and height by y.

c) interpretation of b₀ : For one inch increase in the diameter at breast height, the height of the tree increases by 2.67 feet.

interpretation of b1 : When the diameter at breast height is 0 inch, the tree is 78.79 feet high.

d) For x= 25 inch, y= 78.79+2.67*25 =  145.54 feet

  For x= 60 inch, y= 78.79+2.67*60 = 238.99 feet

e) The coeffecient of determination is :

Therefore , 72.7% of the variation in Height of a tree is explained by the linear regression due to diameter at breast

f) The residual plot after calculating the residuals is

So there doesn't exist any pattern in the graph indicating the randomness of the residuals

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