"In Exercises 7-10, 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?
10. r =0.881"

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 do this, square the given correlation coefficient (r). Use the formula: $r^2$. For this problem, r = 0.881, so compute ${0.881}^2$.
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 = 0.776, it means 77.6% of the variation is explained by the regression model.
4. Step 4: Determine the unexplained variation. Subtract r² from 1 to find the proportion of variation that is not explained by the regression line. Use the formula: $1-r^2$. This represents the unexplained variation.
5. Step 5: Summarize the findings. The explained variation (r²) tells us how well the regression model fits the data, while the unexplained variation (1 - r²) indicates the portion of the data's variability that is not captured by the model. This helps assess the model's effectiveness.

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. In this context, a correlation coefficient of 0.881 indicates a strong positive relationship between the variables.

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). For an r of 0.881, r^2 would be approximately 0.776, meaning about 77.6% of the variation in the dependent variable is explained by the independent variable.

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, while unexplained variation is the portion that remains after accounting for the model. In the context of the coefficient of determination, a higher r^2 value indicates that a larger proportion of the variation is explained by the model, suggesting a better fit. Conversely, the unexplained variation represents the data points that do not conform to the predicted values from the regression line.