Question

The paper "Root Dentine Transparency: Age Determination of Human Teeth Using Computerized Densitometric Analysis" (American Journal of Physical Anthropology \([1991]: 25-30\) ) reported on an investigation of methods for age determination based on tooth characteristics. With \(y=\) age (in years) and \(x=\) percentage of root with transparent dentine, a regression analysis for premolars gave \(n=36\), SSResid \(=5987.16\), and \(\mathrm{SSTo}=\) \(17,409.60 .\) Calculate and interpret the values of \(r^{2}\) and \(s_{e}\)

Short answer

The calculated \(r^{2}\) shows how closely the data follow the regression line. The less the value of \(r^{2}\), the better the model fits your data. The calculated \(s_{e}\) defines the typical deviation of points around the line of best fit. Less values of \(s_{e}\) indicate less variation of observed from predicted values. The exact values for \(r^{2}\) and \(s_{e}\) can be found using a calculator.

Step-by-step solution

Calculate \(r^{2}\) (coefficient of determination)

To calculate the coefficient of determination \(r^{2}\) we need to apply the formula:\[ r^{2} = 1 - (SSResid/SSTo) \]In this instance:\[ r^{2} = 1 - (5987.16/17409.60) \]

Compute \(s_{e}\) (standard error of the estimate)

To calculate \(s_{e}\) we need to use the formula:\[ s_{e} = \sqrt{SSResid/(n-2)} \]Where:- n refers to the sample sizeSubstituting the given values:\[ s_{e} = \sqrt{5987.16/(36-2)} \]

Interpret the results

The \(r^{2}\) value indicates how well the regression line approximates the real data points. The closer the \(r^{2}\) value is to 1, the better the regression line fits the data. \(s_{e}\) provides a measure of the standard distance between the predicted and observed responses in a regression model. Lower values are preferred as they indicate less discrepancy between predicted and observed responses.