The radius of the crown in metres and the diameter at breast

The radius of the crown (in metres) and the diameter at breast height (in centimetres) were measured for ten Sitka spruce trees.

dbh (cm)

Crown Radius (m)

X

Y

5.0

0.91

12.7

1.83

7.6

1.22

17.8

1.83

5.1

1.22

15.2

2.44

10.2

1.70

22.9

2.74

20.3

2.65

10.1

1.52

a) Plot the observations to check if the linear relationship is reasonable.

b) Fit a linear regression of Y_i = b₀ + b₁ X_i

c) Test if the linear regression is significant.

d) Compute the correlation coefficient and the coefficient of determination and discuss their meaning.

e) Calculate Standard Error of Estimate and explain its meaning.

f) Calculate 95% and 99% confidence limits for the slope and intercept. Discuss your results.

g) Calculate the average crown radius for all 17.0 cm trees with 95% confidence limits and discuss your results.

h) Is your average radius calculated for 16.0 cm trees significantly greater than 1.5 metres ( = 0.05)?

i) Is your intercept significantly greater than zero? Use = 0.05.

Solution

First plot the Scatter plot

It looks almost linear.

b) Regression line is y = 0.61+0.094x

d) r = 0.9333 and coeff of deter = 0.8710