If the coefficient of determination R-squared = 1.00, then A. the explained variation equals the unexplained variation B. there is no unexplained variation C. the Y-intercept (B0) must equal 0 D. there is no explained variation
The correct answer and explanation is:
Correct Answer: B. there is no unexplained variation
Explanation (300 words)
The coefficient of determination, R-squared (R²), is a statistical measure that represents the proportion of the variance in the dependent variable (Y) that is predictable or explained by the independent variable(s) (X) in a regression model.
R-squared values range from 0 to 1:
- An R² of 0 means that the model explains none of the variation in the response variable.
- An R² of 1.00 means that the model explains 100% of the variation in the response variable.
Why Option B is Correct
If R² = 1.00, it implies perfect prediction by the regression model. This means:
- Every data point lies exactly on the regression line.
- There is no error (or residual) between the observed Y values and the predicted Y values.
- As a result, the unexplained variation (residual sum of squares) is zero.
Thus, all variation in Y is explained by X, and there is no unexplained variation, making option B the correct choice.
Why the Other Options Are Incorrect
- A. “The explained variation equals the unexplained variation”
Incorrect. This would be true if R² = 0.5, not 1.0. If R² = 1.00, all variation is explained, and unexplained variation is zero. - C. “The Y-intercept (B₀) must equal 0”
Incorrect. R² = 1.00 does not require the intercept to be 0. A model can perfectly fit the data and still have a non-zero intercept. - D. “There is no explained variation”
Incorrect. The opposite is true. When R² = 1.00, all variation is explained.
Conclusion
A perfect R-squared value of 1.00 signifies a flawless fit of the regression model to the data, meaning there is no unexplained variation, making option B the correct answer.