How Much Wood Could a Woodchuck Chuck? By: Lindsey Harrison
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1 How Much Wood Could a Woodchuck Chuck? By: Lindsey Harrison Objective: We will first explore fitting a given data set with various functions in search of the line of best fit. To determine which fit is the best we will compute the sum of squares for error. We will then use this function to predict those data points that are incomplete in the original file. Data and Scatterplot This data is from the lumber industry, giving the approximate number of 100s of board feet of lumber per tree in a tree of a given age. Using Excel we can plot the 7 complete data points on a scatterplot (seen below).
2 Using Excel, we are going to investigate fitting various functions to the data in order to interpolate the data to predict the missing data points.
3 Function Exploration Linear Students are most familiar with linear functions so this may be the natural first guess at fitting the data with a known function. Note that the prediction equation is posted in the top right-hand corner of the following graph. To assess the fit of the linear function, we can look at the coefficient of determination or the sum of squares of the errors. The closer the R 2 value is to one, the better the equation for prediction purposes. The linear regression model gives us an R 2 value of This is not a bad value, but we hope with different curves, we can get this value closer to one. The smaller the sum of squares of errors is, the better the approximating function fits the data. For the linear model the sum of squares of error is approximately 1306.
4 Exponential The next curve we will use to approximate the trend in tree data is an exponential regression equation. In the long run, this family of functions would not accurately model the relationship between age of a tree and 100s of board feet since the tree would eventually die; however, given our range of data, it may be useful to use an exponential curve. The exponential regression model gives us an R 2 value of This is a coefficient of determination that is slightly higher than the linear regression coefficient of determination. For the exponential regression model the sum of squares of error is approximately 14,306. This show us that the exponential curve is definitely not the one we will use to describe the given data.
5 Power The next curve we will use to approximate the trend in tree data is a power regression equation. Again the equation is displayed in the top right-hand corner of the following graph. The power regression model gives us an R 2 value of This high of a coefficient of determination is an indicator that this model is a strong candidate to be used for the prediction of our unknown values. For the power regression model the sum of squares of error is approximately 12. This statistical summary shows that the power function does a good job of explaining the relationship between the tree data points.
6 Polynomial The polynomial curve of degree two is the last curve we will investigate in an attempt to find a prediction equation for the tree data. The polynomial regression model gives us an R 2 value of This coefficient of determination for this polynomial regression equation is much higher than our first two curves (linear and exponential); however, it is not as high as the coefficient of determination from the power function. For the polynomial of degree two regression model the sum of squares of error is approximately 8. This statistical summary shows that the polynomial function also does a good job of explaining the relationship between the tree data points.
7 Making Predictions The two statistical quantities we have been using to assess goodness of fit for our functions are sum of squares due to error (SSE) and the coefficient of determination (R 2 ). Note the sum of squares of error measures the total deviation of the response values (100s of board feet) from the fit to the response values (predicted 100s of board feet). The coefficient of determination measures how successful the fit is in explaining variation in the data. If we use the sum of squares due to error to select the best fit, it would be the polynomial function, but if we use the coefficient of determination to select the best fit, it would be the power function. It is for this reason that we have both the polynomial model and power model below to predict the 100s of board feet for trees of age 60, 140, and 180. There is not a significant amount of variation between the two model s predicted quantities which tells us the both could be useful in predicting 100s of board feet from the age of a given tree.
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