Assignment No.11.The tree data below,obtained by Tamra J.Burcar and used with her permission,consist

of 23 observations of X = tree diameter in inches measured at breast height (DBH) and Y =

height in feetX Y5.5 58

5.7 60

6.5 64

6.6 60

6.7 65

6.9 56

7.0 57

7.3 70

8.3 68

8.6 65

9.5 70

10.0 63

10.1 75

10.2 72

10.4 78

10.6 65

10.6 80

10.8 82

11.3 70

11.3 74

11.6 68

11.6 68

13.0 82(a) Plot Y versus X.

(b) Fit Y =

0

+

1

X + by least squares and plot the tted line on your diagram.

(c) Evaluate the residuals to two decimal places.Check that

P

n

i=1

e

i

= 0;within rounding

error.

(d) Obtain the full analysis of variance table in the form given in ARA (shorthand for

Applied Regression Analysistext),p.31,Table 1.4.

(e) Find out how much of the variation about the mean

Y is explained by the tted line.

(f) Use the repeat runs to test for lack of t and state your conclusions.

(g) Is it appropriate to test for overall regression via the F-test?If so,do it.

(h) nd standard errors for b

0

and b

1

;using s

2

e

or s

2

as seems appropriate from (f).

(i) Find the formula for the standard error of

^

Y and,assuming that s

2

is an appropriate

estimate of

2

for this part,construct 95% condence bands for the true mean value of Y:

(Plot about half a dozen points over the X-range and join them up smoothly.)

1

(In fact two more data points were obtained by Ms.Burcar,but were deleted from the

data by your instructor.They wereX Y5.8 42

18.0 88The small tree was clearly stunted,the large tree may have been damaged by winds

because it protruded above the tree line.)

(j) Show the new trees on your plot and comment brie y on what you see.

2.Reconsider the data in problem 1.

a.Fit the model X =

0

+

1

Y + by least squares.

b.Find the tted line that passes through (

X;

Y ) and has slope

c

1

= (b

1

=a

1

)

1=2

;

where b

1

is the least squares estimate of

1

in the Y =

0

+

1

X + t of question 1 and

a

1

is the least squares estimate of

1

in the model of this question 2.This tted line is the

\geometric mean functional relationship",as described by F.Barker,Y.C.Soh,and R.J.

Evans,in\Properties of the geometrical mean functional relationship",Biometrics,44,1988,

279-281.

c.State exactly what this third line is in the form

^

Y = c

0

+c

1

X;and plot all three lines

on a new diagram.

2

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