 Mathematics Explorations with Microsoft Excel Spreadsheet

Assignment 12

By Gloria L. Jones

This assignment will demonstrate how Microsoft Excel (spreadsheets) can be used to manipulate data graphically, in particular, create mathematical models for data to be used for regression analysis.

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

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

Typically when one first obtains data, a scatter plot should be generated to detect possible signs of a mathematical relationship: At a glance one might guess that there exists a mathematical model to approximate this data.  The shape of the graph hints at possibly an exponential or power function to better model the data.

Letπs first attempt to fit an exponential curve to the data: I

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).   Note that the R² value is 0.8968.  The higher this value is the better the prediction.

Also, notice the residuals (difference between observed and predicted values) for this model: It is clear that the exceptionally large residual at age 200 which contributes strongly to an average sum of squares of 14306.15646.

Indicators hint that a power regression model would probably serve the data better: Notice how exceptionally tight the trend line fits the data points with considerably lower residual error.  Most likely this will serve as our best model for the data.  The R² value of 0.9999 also serves as a quick determinant of how close the fit is.  Note that the power function has a higher R². The mean sum of squares is a relatively low 11.71996878.

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 out put in 100s of board feet are 16.232, 134.15, and 250.983 respectively.