"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?
15. r = 0.642"

Verified step by step guidance

1. Step 1: Recall the formula for the coefficient of determination, r², which is the square of the correlation coefficient r. Mathematically, r² = r × r.
2. Step 2: Substitute the given value of r (0.642) into the formula. This means you will calculate r² = (0.642)².
3. Step 3: Interpret the coefficient of determination, r². It represents the proportion of the variation in the dependent variable (y) that is explained by the independent variable (x) through the regression line.
4. Step 4: To find the explained variation, multiply r² by 100 to express it as a percentage. This percentage indicates how much of the total variation in the data is explained by the regression model.
5. Step 5: To find the unexplained variation, subtract the explained variation percentage from 100%. This represents the proportion of the variation in the data that is not explained by the regression model.

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 indicate a strong positive correlation, values close to -1 indicate a strong negative correlation, and values around 0 suggest no linear correlation. Understanding r is crucial for interpreting the relationship between the variables in a dataset.

Coefficient of Determination (r^2)

The coefficient of determination, represented as r^2, quantifies the proportion of variance in the dependent variable that can be explained by the independent variable in a regression model. It is calculated by squaring the correlation coefficient (r). An r^2 value closer to 1 indicates that a large proportion of the variance is explained by the model, while a value closer to 0 suggests that the model explains little of the variance.

Explained vs. Unexplained Variation

Explained variation refers to the portion of the total variation in the dependent variable that is accounted for by the regression model, as indicated by r^2. Conversely, unexplained variation is the portion that remains after accounting for the model, representing the variability that cannot be predicted by the independent variable. Understanding these concepts helps in assessing the effectiveness of the regression model in capturing the underlying data patterns.