Answer: The coefficient estimate $ \beta_4 $ is non-zero but not significant

The fact that R-squared is higher for model 2 (0.81 vs 0.75) but the adjusted R-squared is lower for model 2 (0.72 vs 0.73) suggests that the extra variable $ x_4 $ is not statistically significant. **Key points:** 1. **R-squared always increases** when adding more variables, even if they are irrelevant (0.81 > 0.75). 2. **Adjusted R-squared** penalizes for adding insignificant variables. The decrease from 0.73 to 0.72 indicates $ x_4 $ doesn't contribute enough explanatory power to justify its inclusion. 3. **Option A is incorrect** because if $ \beta_4 $ was exactly zero, R-squared would not increase. 4. **Option B is incorrect** because adjusted R-squared can decrease when adding insignificant variables. 5. **Option C is incorrect** because if $ x_4 $ were statistically significant, adjusted R-squared would typically increase. 6. **Option D is correct** because the pattern suggests $ \beta_4 $ is non-zero (R-squared increased) but not statistically significant (adjusted R-squared decreased).

Author: Nikitesh Somanthe