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.

Weight , Waist r = 0.885 (x = weight of male, y = waist size of male)

Short answer

The coefficient of determination is 0.783.

It means 78.3% of the variation is explained by the linear association between the weight and the waist size of a male.

And 21.7% of the variation in the response variable (waist size of male)is explained by other factors and random variation.

Step-by-step solution

Given information

The linear correlation coefficient between the weight and the waist size of a male is 0.885.

Coefficient of determination

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

Here, the linear correlation coefficient (r) between the weight and the waist size of a male is 0.885.

Thus,

\(\begin{array}{c}{\rm{Coefficient}}\;{\rm{of}}\;{\rm{determination}} = {r^2}\\ = {0.885^2}\\ = 0.783\end{array}\)

Therefore, the value of the coefficient of determination is 0.783.

Percentage of variation

Here,

\(\begin{array}{c}{r^2} = 0.783\\ = \frac{{78.3}}{{100}} \times 100\% \\ = 78.3\% \end{array}\)

Therefore, the percentage of the variation explained by the linear association between the weight and the waist size of males is 98.4%.

The rest \(100\% - 78.3\% = 21.7\% \) variation is explained by other factors and random variation.