Question
In: Statistics and Probability
Measuring the height of a California redwood tree is very difficult because these trees grow to...
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 a
California
redwood tree is related to other characteristics of the tree,
including the diameter of the tree at the breast height of a
person.
The data in Redwood represent the height (in feet) and
diameter
(in inches) at the breast height of a person for a sample of 21
California
redwood trees.
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 25 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 model.
Data Is Below:
| Height | Diameter at breast height | Bark thickness |
| 122.0 | 20 | 1.1 |
| 193.5 | 36 | 2.8 |
| 166.5 | 18 | 2.0 |
| 82.0 | 10 | 1.2 |
| 133.5 | 21 | 2.0 |
| 156.0 | 29 | 1.4 |
| 172.5 | 51 | 1.8 |
| 81.0 | 11 | 1.1 |
| 148.0 | 26 | 2.5 |
| 113.0 | 12 | 1.5 |
| 84.0 | 13 | 1.4 |
| 164.0 | 40 | 2.3 |
| 203.3 | 52 | 2.0 |
| 174.0 | 30 | 2.5 |
| 159.0 | 22 | 3.0 |
| 205.0 | 42 | 2.6 |
| 223.5 | 45 | 4.3 |
| 195.0 | 54 | 4.0 |
| 232.5 | 39 | 2.2 |
| 190.5 | 36 | 3.5 |
| 100.0 | 8 | 1.4 |
Solutions
Expert Solution
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.
The regression equation is:
Height = 78.7911 +
2.6735*diameter at breast height
b. Interpret the meaning of the slope in this equation.
For every diameter at breast height, height of a tree increases
by 2.6735.
c. Predict the mean height for a tree that has a breast height
diameter
of 25 inches.
Height = 145.6280
d. Interpret the meaning of the coefficient of determination in
this
problem.
72.9% of the variation in the model is explained.
e. Perform a residual analysis on the results and determine the
adequacy
of the model.
The residual plot is:
The model is adequate.
| Height | Diameter at breast height | |||||
| 122 | 20 | |||||
| 193.5 | 36 | |||||
| 166.5 | 18 | |||||
| 82 | 10 | |||||
| 133.5 | 21 | |||||
| 156 | 29 | |||||
| 172.5 | 51 | |||||
| 81 | 11 | |||||
| 148 | 26 | |||||
| 113 | 12 | |||||
| 84 | 13 | |||||
| 164 | 40 | |||||
| 203.3 | 52 | |||||
| 174 | 30 | |||||
| 159 | 22 | |||||
| 205 | 42 | |||||
| 223.5 | 45 | |||||
| 195 | 54 | |||||
| 232.5 | 39 | |||||
| 190.5 | 36 | |||||
| 100 | 8 | |||||
| r² | 0.729 | |||||
| r | 0.854 | |||||
| Std. Error | 24.747 | |||||
| n | 21 | |||||
| k | 1 | |||||
| Dep. Var. | Height | |||||
| ANOVA table | ||||||
| Source | SS | df | MS | F | p-value | |
| Regression | 31,279.339 | 1 | 31,279.339 | 51.08 | 8.58E-07 | |
| Residual | 11,635.946 | 19 | 612.4182 | |||
| Total | 42,915.286 | 20 | ||||
| Regression output | confidence interval | |||||
| variables | coefficients | std. error | t (df=19) | p-value | 95% lower | 95% upper |
| Intercept | 78.7911 | |||||
| Diameter at breast height | 2.6735 | 0.3741 | 7.147 | 8.58E-07 | 1.8905 | 3.4564 |
| Observation | Height | Predicted | Residual | |||
| 1 | 122.00 | 132.26 | -10.26 | |||
| 2 | 193.50 | 175.04 | 18.46 | |||
| 3 | 166.50 | 126.91 | 39.59 | |||
| 4 | 82.00 | 105.53 | -23.53 | |||
| 5 | 133.50 | 134.93 | -1.43 | |||
| 6 | 156.00 | 156.32 | -0.32 | |||
| 7 | 172.50 | 215.14 | -42.64 | |||
| 8 | 81.00 | 108.20 | -27.20 | |||
| 9 | 148.00 | 148.30 | -0.30 | |||
| 10 | 113.00 | 110.87 | 2.13 | |||
| 11 | 84.00 | 113.55 | -29.55 | |||
| 12 | 164.00 | 185.73 | -21.73 | |||
| 13 | 203.30 | 217.81 | -14.51 | |||
| 14 | 174.00 | 159.00 | 15.00 | |||
| 15 |
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