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.

Crickets and Temperature r = 0.874 (x = number of cricket chirps in 1 minute, y = temperature in °F)

Short answer

The coefficient of determination is 0.764.

The percentage of variation that can be explained by the linear association between the number of cricket chirps in 1 minute and the temperature is 76.4%.

23.6% of the variation in the response variable (temperature) is explained by other factors and random variation.

Step-by-step solution

Given information

The linear correlation coefficient between the number of cricket chirps in 1 min and the temperature is 0.874.

Coefficient of determination

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

Here, the linear correlation coefficient (r) between the number of cricket chirps in 1 minute and the temperature is 0.874.

Thus,

\(\begin{array}{c}{\rm{Coefficient}}\;{\rm{of}}\;{\rm{determination}} = {r^2}\\ = {0.874^2}\\ = 0.764\end{array}\)

Therefore, the value of the coefficient of determination is 0.764.

Percentage of variation

Here,

\(\begin{array}{c}{r^2} = 0.764\\ = \frac{{76.4}}{{100}} \times 100\% \\ = 76.4\% \end{array}\)

Therefore, the percentage of the variation explained by the linear association of the number of cricket chirps in 1 min and the temperature is 76.4%.

The remaining \(100\% - 76.4\% = 23.6\% \) variation is explained by other factors and random variation.