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. (Hand, 1994)

	Estimate	Std. Error	t value	P(>|t|)
(Intercept)	-57.99	8.64	-6.71	0.00
height	0.34	0.13	2.61	0.01
diameter	4.71	0.26	17.82	0.00

(a) Write the regression model (please do not round coefficients):
E[y]=E[y]=  + ⋅height+⋅height+ ⋅diameter⋅diameter

(b) Keeping diameter constant, how much additional volume should we expect from an average tree if height is increased by 1 foot?
cubic feet

(c) Are each of the predictors, "height" and "diameter" significant predictors of volume?

• Yes, since the p-values associated with each predictor are less than 0.05
• No, since the p-values associated with each predictor are less than 0.05
• Only diameter is a significant predictor since it has the smallest p-value

(d) How much volume is expected from a tree that measures 79 feet tall and has a diameter of 11.3 inches? (please round to the nearest cubic foot)
cubic feet

(e) A tree in the data set measures 79 feet tall, has a diameter of 11.3 inches, and is 24.2 cubic feet in volume. Determine whether the model gives an overestimate or underestimate of the volume of this tree.

• underestimate
• overestimate

Answer 1

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