Which of the following is the correct interpretation for 0.60 coefficient of determination?

Question

BlackTom AI · Accepted Answer

Coefficient of determination R^2 = 0.60 means 60% of the variance in the dependent variable Y is accounted for by the model (the variation in X).

1. Relevant knowledge points and concepts
- R^2 (coefficient of determination) measures the proportion of the total variation in the response variable (Y) explained by the explanatory variable(s) (X) and the fitted model.
- In simple linear regression, R^2 = (correlation coefficient r)^2, so R^2 ranges from 0 to 1 and is interpreted as a percentage when multiplied by 100.
- 1 − R^2 is the proportion of variance in Y not explained by X (error or unexplained variance).

2. Step-by-step reasoning
- Given R^2 = 0.60, convert to a percentage: 0.60 × 100% = 60%.
- That 60% refers to the proportion of Y’s variation explained by X (by definition of R^2).
- The unexplained portion is 1 − 0.60 = 0.40, or 40% of Y’s variation remains unexplained by X.

3. Calculations (intermediate steps)
- R^2 = 0.60
- Explained variance = 0.60 × 100% = 60%
- Unexplained variance = (1 − 0.60) × 100% = 0.40 × 100% = 40%
- If desired, correlation magnitude in simple linear regression: |r| = sqrt(0.60) ≈ 0.7746 (shows a moderately strong linear association).

4. Why the correct answer is correct
- By definition, R^2 quantifies the fraction of Y’s total variance explained by X. Thus “60% of the variation in the Y values is explained by the variation in the X values” is the precise interpretation.

5. Why the other options are incorrect
- Option A (“60% of the variation in the X values is explained by the variation in the Y values”): reverses the dependent/independent roles; R^2 pertains to Y explained by X, not vice versa.
- Option B (“The relationship between Y and X is very weak”): R^2 = 0.60 indicates a moderate-to-strong relationship, not “very weak.”
- Option D (“40% of the variation in the Y values is explained by the variation in the X values”): 40% is the unexplained portion; the explained portion is 60%.

6. Conclusion
This confirms the correct interpretation is: “60% of the variation in the Y values is explained by the variation in the X values.”

Which of the following is the correct interpretation for 0.60 coefficient of determination?单项选择题

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