"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?
16. r = 0.795"

Verified step by step guidance

1. Identify the given correlation coefficient, which is $r=0.795$.
2. Calculate the coefficient of determination by squaring the correlation coefficient: $r^2=\; (0.795)^2$. This value is denoted as $r^2$.
3. Interpret $r^2$ as the proportion of the total variation in the dependent variable that is explained by the variation in the independent variable through the regression line.
4. Understand that the explained variation corresponds to $r^2$, while the unexplained variation corresponds to $1\; -\; r^2$, representing the variation not accounted for by the regression model.
5. Summarize that a higher $r^2$ value indicates a better fit of the regression line to the data, meaning more of the variation is explained, and less is unexplained.

Key Concepts

Correlation Coefficient (r)

The correlation coefficient, denoted as r, measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where values close to 1 or -1 indicate a strong linear relationship, and values near 0 indicate little to no linear association.

Coefficient of Determination (r²)

The coefficient of determination, r², is the square of the correlation coefficient and represents the proportion of the variance in the dependent variable explained by the independent variable. It ranges from 0 to 1, with higher values indicating a better fit of the regression line to the data.

Explained vs. Unexplained Variation

Explained variation refers to the portion of total variation in the data accounted for by the regression model, quantified by r². Unexplained variation is the remaining variation not captured by the model, representing random error or other factors outside the regression.