Question

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)

Short answer

The coefficient of determination is 0.613.

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

And 38.7% variation in the weight of a bear is explained by other factors and random variation.

Step-by-step solution

Given information

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

Coefficient of determination

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.

Percentage of variation

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.