Answer: $$\hat{Y} = -1.0799 + 0.5633X_1$$

## Explanation The correct answer is **B** because the estimated regression equation matches the calculated parameters from the given data. ### Step-by-Step Calculation: 1. **Calculate Covariance between Y and X₁:** $$\text{Cov}(Y, X_1) = \frac{1}{n - 1} \sum_{i=1}^{n} (X_1 - \bar{X}_1)(Y - \bar{Y})$$ From the table, the total sum is 2.759533 $$\text{Cov}(Y, X_1) = \frac{1}{6 - 1} \times 2.759533 = 0.5519$$ 2. **Calculate Variance of X₁:** $$\text{Var}(X_1) = \frac{1}{n - 1} \sum_{i=1}^{n} (X_1 - \bar{X}_1)^2 = 0.9797$$ 3. **Calculate Slope Coefficient (β₁):** $$\hat{\beta}_1 = \frac{\text{Cov}(Y, X_1)}{\text{Var}(X_1)} = \frac{0.5519}{0.9797} = 0.5633$$ 4. **Calculate Intercept (α):** $$\hat{\alpha} = \bar{Y} - \hat{\beta}_1 \bar{X}_1$$ $$\hat{\alpha} = -0.82833 - (0.5633 \times 0.446667) = -1.0799$$ Therefore, the estimated regression equation is: $$\hat{Y} = -1.0799 + 0.5633X_1$$ This matches option **B** exactly.

Author: Tanishq Prabhu