Coefficient of Determination Using the heights and weights described in Exercise 1, the linear correlation coefficient r is 0.394. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

Verified step by step guidance

1. The coefficient of determination, denoted as R², is calculated by squaring the linear correlation coefficient r. The formula is: $R^2=r^2.$
2. Substitute the given value of r (0.394) into the formula: $R^2={0.394}^2.$
3. Simplify the expression by squaring 0.394 to find the value of R².
4. The coefficient of determination, R², represents the proportion of the variance in the dependent variable (e.g., weight) that is predictable from the independent variable (e.g., height).
5. In practical terms, the value of R² indicates how well the linear regression model explains the variability of the dependent variable. A higher R² value means a better fit of the model to the data.

Key Concepts

Coefficient of Determination (R²)

The coefficient of determination, denoted as R², quantifies the proportion of variance in the dependent variable that can be explained by the independent variable(s) in a regression model. It ranges from 0 to 1, where 0 indicates no explanatory power and 1 indicates perfect explanation. In practical terms, a higher R² value suggests a better fit of the model to the data.

Linear Correlation Coefficient (r)

The linear correlation coefficient, represented 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. The square of r (r²) is used to calculate the coefficient of determination.

Practical Interpretation of R²

The practical interpretation of the coefficient of determination (R²) provides insights into how well the independent variable(s) predict the dependent variable. For instance, if R² is 0.155, it indicates that approximately 15.5% of the variance in the dependent variable can be explained by the independent variable. This information is crucial for assessing the effectiveness of the model in real-world applications.