The data in the following table represents height (in meters) and the diameter (
ID: 3052629 • Letter: T
Question
The data in the following table represents height (in meters) and the diameter (in cms) of a specific species of trees.
Table 1: height and diameters of trees
Specimen
Diameter
Height
1
32
22.7
2
31
22.7
3
30
22.6
4
29
22.6
5
29
21.9
6
28
21.9
7
25
21.8
8
23
21
9
20
20.4
10
18
18.6
11
17
19.2
12
17
18.9
13
16
18.5
14
16
18.1
15
15
17.7
16
13
17.2
17
11
16.5
18
11
15.5
We would like to check if we can express the tree height as a function of the diameter.
1. We look at linear regression first.
a. Show all the steps that you perform to find the coefficients “manually”.
b. Is this model a good fit? Compute R2 and plot the standardized residuals plotted against the fitted values to support your conclusion.
2. Higher order regression.
a. Use your favorite software to perform quadratic and cubic regression (which means by trying to fit the height as a polynomial of degree two or three of the diameter).
b. Are these models good fits?
c. Is the cubic regression improving a lot with respect to the quadratic one?
3. Actually an alternative modeling is to explore the possibility of an exponential relation between height and diameter. This means that we want to explore the relation h=a*db where h is the height, d is the diameter and a and b are coefficients.
a. What happens to the previous relation if we take the logarithm of both sides?
b. Perform a linear regression on the log scale (ln(h) and ln(d))
c. Derive the corresponding relation on the original parameters h and d. How do you interpret the corresponding result?
4. Compute 95% prediction intervals for the diameters 11, 20 and 30 cm in all four models (linear, quadratic, cubic and log). What conclusion can you draw?
Specimen
Diameter
Height
1
32
22.7
2
31
22.7
3
30
22.6
4
29
22.6
5
29
21.9
6
28
21.9
7
25
21.8
8
23
21
9
20
20.4
10
18
18.6
11
17
19.2
12
17
18.9
13
16
18.5
14
16
18.1
15
15
17.7
16
13
17.2
17
11
16.5
18
11
15.5
Explanation / Answer
-Solution:-
1.a. Let us consider, Diameter=xi and Hight=yi
Since we have to find linear regression equation : ? = b0 + b1x ,where b1=regression coefficient,b0=slope
b1 = ? [ (xi - x?)(yi - ?) ] / ? [ (xi-x?)^2]
=283.56/890.5=0.318
Now ,we have the value of the regression coefficient (b1), we can solve for the regression slope (b0):
b0 = ? - b1 * x?=19.87-(21.16*.318)=13.14
Therefore regression equation=? = b0 + b1x =13.14+0.318x
b.Ans:--
Calculate R2 which is Coefficient of Determination,higher the value of R2 the more the value is consdered as good fit.
R2 = { ( 1 / N ) * ? [ (xi - x?) * (yi - ?) ] / (?x * ?y ) }2, Where where N is the number of observations used to fit the model,?x is the standard deviation of x, and ?y is the standard deviation of y.
?x = sqrt [ ? ( xi - x )2 / N ]=7.03, ?y = sqrt [ ? ( yi - y )2 / N ]=2.29
R2={ ( 1 / 18 ) * [ 283.56 ] / (7.03*2.29)}2 =(17.61/18)2 =.978 =.98 apporx.=98%
98% of the variation in height and diameter o the tree can be explained by the relationship. This would be considered a good fit to the data
Specimen xi yi xi-x? yi -? (xi-x?)^2 (yi -?)^2 (xi-x?)(yi -?) 1 32 22.7 10.833333 2.8222222 117.36111 7.9649383 30.574074 2 31 22.7 9.8333333 2.8222222 96.694444 7.9649383 27.751852 3 30 22.6 8.8333333 2.7222222 78.027778 7.4104938 24.046296 4 29 22.6 7.8333333 2.7222222 61.361111 7.4104938 21.324074 5 29 21.9 7.8333333 2.0222222 61.361111 4.0893827 15.840741 6 28 21.9 6.8333333 2.0222222 46.694444 4.0893827 13.818518 7 25 21.8 3.8333333 1.9222222 14.694444 3.6949383 7.3685185 8 23 21 1.8333333 1.1222222 3.3611111 1.2593827 2.0574074 9 20 20.4 -1.1666667 0.5222222 1.3611111 0.272716 -0.6092593 10 18 18.6 -3.1666667 -1.2777778 10.027778 1.6327161 4.0462963 11 17 19.2 -4.1666667 -0.6777778 17.361111 0.4593827 2.8240741 12 17 18.9 -4.1666667 -0.9777778 17.361111 0.9560494 4.0740741 13 16 18.5 -5.1666667 -1.3777778 26.694444 1.8982716 7.1185185 14 16 18.1 -5.1666667 -1.7777778 26.694444 3.1604938 9.1851852 15 15 17.7 -6.1666667 -2.1777778 38.027778 4.7427161 13.42963 16 13 17.2 -8.1666667 -2.6777778 66.694444 7.1704938 21.868519 17 11 16.5 -10.166667 -3.3777778 103.36111 11.409383 34.340741 18 11 15.5 -10.166667 -4.3777778 103.36111 19.164938 44.507407 SUM 381 357.8 890.5 94.75 283.56 MEAN x?=21.16667 ?=19.87778Related Questions
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