8.6 Cherry trees. Timber yield is approximately equal to the volume of a tree, h

Question

8.6 Cherry trees. Timber yield is approximately equal to the volume of a tree, however, this value is difficult to measure without first cutting the tree down. Instead, other variables, such as height and diameter, may be used to predict a tree's volume and yield. Researchers wanting to understand the relationship between these variables for black cherry trees collected data from 31 such trees in the Allegheny National Forest, Pennsylvania. Height is measured in feet, diameter in inches (at 54 inches above ground), and volume in cubic feet.19 Estimate Std. Error t value Prt (Intercept) height diameter -57.99 0.34 4.71 8.64 0.13 0.26 17.82 6.71 2.61 0.00 0.01 0.00 (a) Calculate a 95% confidence interval for the coefficient of height, and interpret it in the context of the data (b) One tree in this sample is 79 feet tal, has a diameter of 11.3 inches, and is 24.2 cubic feet in volume. Determine if the model overestimates or underestimates the volume of this tree, and by how much

Explanation / Answer

Result:

a).

df for t = 28

critical t at 95% =2.048

CI= ±t*se

95% CI for height = (0.34-2.048*0.13, 0.34+2.048*0.13)

=(0.07376, 0.60624)

We are 95% confident that when height increases by 1 feet, average volume increases falls in the interval (0.07376, 0.60624).

b).

The regression line is

volume =-57.99+0.34*height+4.71*diameter

predicted volume =-57.99+0.34*79+4.71*11.3

=22.093

The model under estimates by 24.2-22.093 =2.107

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