Question

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. r = 0.437 what is the value of the coefficient of determination? $r^2 = \square$ (round to four decimal places as needed.) what is the percentage of the total variation that can be explained by the linear relationship between the two variables? explained variation = \square \\% (round to two decimal places as needed.)

Explanation:

Step1: Calculate coefficient of determination

The coefficient of determination \( r^2 \) is the square of the linear correlation coefficient \( r \). Given \( r = 0.437 \), we calculate \( r^2 \) as \( 0.437^2 \).
\( 0.437 \times 0.437 = 0.190969 \), rounded to four decimal places is \( 0.1910 \).

Step2: Calculate explained variation percentage

To find the percentage of explained variation, we multiply \( r^2 \) by 100. So, \( 0.190969 \times 100 = 19.0969\% \), rounded to two decimal places is \( 19.10\% \).

Answer:

Coefficient of determination: \( \boldsymbol{0.1910} \)
Explained variation: \( \boldsymbol{19.10\%} \)