Assignment 12

By: Sharren M. Thomas

The problem:

The following data represents lumber output per tree (in 100s of board feet) based on the age of the tree:

 Age of Tree 100s of Board Feet 20 1 40 6 60 80 33 100 56 120 88 140 160 182 180 200 320

What function will fit the data?  Predict the harvest for ages other than those given.  See below the scatter plot for the given data.

I will try to approximate this data.  The shape of the graph looks as if it is a power function or exponential curve.

I will try to fit an exponential curve to the data:

The residuals (difference between observed and predicted values). The curve fits closely for the lower numbers but as the age of the tree increases, the observed data does not increase at such a high rate ( notice the outlier at age 200).  The residuals  for this model:

 Age of Tree 100s of Board Feet Residuals 20 1 -1.862 40 6 0.78509 60 80 33 15.682 100 56 24.44 120 88 30.488 140 160 182 -8.993 180 200 320 -314.3

There is a large residual at age 200 which contributes strongly to an average sum of squares of 14377.45776.

A power regression model may be better:

This is the best model for the data.

 Age of Tree 100s of Board Feet Residuals 20 1 -0.026 40 6 0.22586 60 80 33 0.50342 100 56 -0.6756 120 88 -1.282 140 160 182 -0.889 180 200 320 1.0326

The mean sum of squares is a very low 0.6088090794.

Being careful not to extrapolate beyond our range of values we can now use the power model equation

y = 0.0006x^(2.4926)

to predict the lumber harvest for trees that are 60, 140 and 180 years old.  The lumber output in 100s of board feet are 15.864, 131.11 and 245.296 respectively.