The data below came from the Allegheny National Forest in Pennsylvania. The issue was whether either the diameter or the height of a tree accurately predict the volume of wood in the tree. Using the data below, you are to see if there exists a meaningful relation between these variables. Thus, you want to find the volume as a function of either diameter or height. The volume is measured in board feet.

Diameter	Height	Volume
8.6	65	10.3
10.7	81	18.8
11.0	75	18.2
11.4	76	21.4
12.0	75	19.1
13.3	86	27.4
14.5	74	36.3
16.0	72	38.3
17.3	81	55.4
18.0	80	51.5
20.6	87	77.0

 

a. The reference for this model suggests a simple linear model, so use Excel's trendline to find the best line through the data. Graph the data and model for volume as a function of diameter, then repeat the process for volume as a function of height. Give the formula for the best straight line through each of the data sets. Which graph seems to have the better predictive ability? Why is this what you would expect based on the biology of trees? What happens with both models as the diameter or height gets close to zero?

b. In this part of the problem only use the relationship between volume and the one variable that you showed in Part a. was the better predictor. Use the power law under Excel's trendline to best fit the one set of data that best predicts the volume. Plot the data and the best power law fit, then have Excel write the formula on your graph. Can you provide any explanation for the power that you have obtained?

c. The concept we use to fit power law to data is fitting a straight line to the logarithms of the data. Thus, from an allometric model of the form y = kx^a , we obtain the formula ln(y) = ln(k) + a ln(x). With the best fitting model from Part b, take the logarithm of the volume and the logarithm of the either the diameter or height (whichever gives the better model). Use Excel's scatter plot and linear fit under trendline to see how this fits these data. Plot a graph of the logarithm of the data and the best straight line fit to these data. Show the formula for this straight line on your graph. Compare the coefficients obtained in this manner to the ones found in Part b. How well does the graph match the data?

d. Have Excel plot a log-log plot of the data and the trendline that you found in Part b. (This is done by editing the graph and selecting logarithmic scales for both the x and y axes, which is easily done by double clicking on the axes.) Do the data roughly fall on a straight line in this plot?