"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?
13. r =- 0.450"

Verified step by step guidance

1. Step 1: Understand the problem. The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. The coefficient of determination (r²) quantifies the proportion of the variation in the dependent variable that is explained by the independent variable in the regression model.
2. Step 2: Calculate the coefficient of determination (r²). To find r², square the given correlation coefficient (r). For this problem, r = -0.450, so r² = (-0.450)². Use the formula: $r^2={\left(}^r.$
3. Step 3: Interpret the coefficient of determination (r²). The value of r² represents the proportion of the total variation in the dependent variable that is explained by the regression line. For example, if r² = 0.2025 (after squaring -0.450), this means 20.25% of the variation is explained by the regression model.
4. Step 4: Determine the unexplained variation. The unexplained variation is the remaining proportion of the total variation that is not explained by the regression line. This is calculated as 1 - r². For example, if r² = 0.2025, then the unexplained variation is 1 - 0.2025 = 0.7975, or 79.75%.
5. Step 5: Summarize the findings. The coefficient of determination (r²) helps us understand how well the regression model fits the data. A higher r² value indicates a better fit, meaning more of the variation is explained by the model. Conversely, a lower r² value indicates that a larger proportion of the variation 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 -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation. In this context, a negative r value suggests that as one variable increases, the other tends to decrease.

Coefficient of Determination (r^2)

The coefficient of determination, r^2, is the square of the correlation coefficient and represents the proportion of variance in the dependent variable that can be explained by the independent variable in a regression model. It ranges from 0 to 1, where a value closer to 1 indicates that a large proportion of the variance is explained by the model, while a value closer to 0 indicates little explanatory power.

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 factors not captured by the independent variable. Understanding these concepts helps in assessing the effectiveness of the regression model in predicting outcomes.