Interpreting the Coefficient of Determination. In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Bears r = 0.783 (x = head width of a bear, y = weight of a bear)

The linear correlation coefficient between the head width and the weight of a bear is 0.783.

The square of the linear correlation coefficient between the two variables equals the coefficient of determination.

Here, the linear correlation coefficient (r) between the head width and the weight of a bear equals0.783.

Thus,

\(\begin{array}{c}{\rm{Coefficient}}\;{\rm{of}}\;{\rm{determination}} = {r^2}\\ = {0.783^2}\\ = 0.613\end{array}\)

Therefore, the value of the coefficient of determination is 0.613.

Here,

\(\begin{array}{c}{r^2} = 0.613\\ = \frac{{61.3}}{{100}} \times 100\% \\ = 61.3\% \end{array}\)

Therefore, the percentage of the variation explained by the linear association between the head width and the weight of a bear is 61.3%.

The rest \(100\% - 61.3\% = 38.7\% \) variation is explained by other factors and random variation.