Questions: Use the value of the linear correlation coefficient to calculate the coefficient of determination. What does this tell you about the explained variation of the data about the regression line? About the unexplained variation? r=-0.813 Calculate the coefficient of determination. 0.661 (Round to three decimal places as needed.) What does this tell you about the explained variation of the data about the regression line? 0.13% of the variation can be explained by the regression line. (Round to one decimal place as needed.)

Solution Steps

To calculate the coefficient of determination, we need to square the linear correlation coefficient \( r \). The coefficient of determination, denoted as \( R^2 \), represents the proportion of the variance in the dependent variable that is predictable from the independent variable.

1. Square the given linear correlation coefficient \( r \) to find \( R^2 \).
2. Interpret \( R^2 \) to understand the explained variation.
3. Calculate the unexplained variation as \( 1 - R^2 \).

Given the linear correlation coefficient \( r = -0.813 \), we calculate the coefficient of determination \( R^2 \) by squaring \( r \): \[ R^2 = (-0.813)^2 = 0.6610 \]

The coefficient of determination \( R^2 \) represents the proportion of the variance in the dependent variable that is predictable from the independent variable. Therefore, the explained variation is: \[ \text{Explained Variation} = R^2 \times 100\% = 0.6610 \times 100\% = 66.10\% \]

The unexplained variation is the remaining proportion of the variance that is not explained by the regression line: \[ \text{Unexplained Variation} = (1 - R^2) \times 100\% = (1 - 0.6610) \times 100\% = 33.90\% \]

Final Answer