Answer: An increase in the R² or $ar{R}^2$ always means that an added variable is statistically significant

## Explanation Statement A is incorrect because: 1. **R² always increases when a regressor is added** - This is a mathematical property of R². Adding any variable (even a random one) will increase or at least not decrease R². 2. **$ar{R}^2$ (adjusted R²) does not always increase** - Unlike R², adjusted R² penalizes for adding variables that don't improve the model. It only increases if the added variable improves the model more than would be expected by chance. 3. **Statistical significance requires a t-test** - Even when R² or $ar{R}^2$ increases, this doesn't guarantee the added variable is statistically significant. To determine significance, we need to perform a t-test on the coefficient. **Why the other statements are correct:** - **Statement B**: High R² doesn't imply causality. Correlation ≠ causation. - **Statement C**: High R² doesn't mean we have the best set of predictors; low R² doesn't mean we have bad predictors. - **Statement D**: High R² doesn't rule out omitted variable bias. We could have high R² but still miss important variables. **Key takeaway**: R² measures goodness of fit, not statistical significance or model adequacy. Always use hypothesis tests (t-tests, F-tests) to evaluate variable significance.

Author: Nikitesh Somanthe