Answer: $\hat{\alpha} = -1.080$, $\hat{\beta}_1 = 0.5633$

## Explanation The correct answer is A: $\hat{\alpha} = -1.080$, $\hat{\beta}_1 = 0.5633$ ### Calculation Process: We need to use the standard formulas for linear regression: $$\hat{\beta}_1 = \frac{\text{Cov}(Y, X_1)}{\text{Var}(X_1)}$$ $$\hat{\alpha} = \bar{Y} - \hat{\beta}_1 \bar{X}_1$$ Where: - $\text{Cov}(Y, X_1) = \frac{1}{(n - 1)} \sum_{i=1}^{n} (Y_i - \bar{Y})(X_{1i} - \bar{X}_1)$ - $\text{Var}(X_1) = \frac{1}{(n - 1)} \sum_{i=1}^{n} (X_{1i} - \bar{X}_1)^2$ **Step 1: Calculate means** - $\bar{Y} = \frac{-2 + (-0.11) + (-1.68) + (-0.36) + (-0.08) + (-0.74)}{6} = \frac{-4.97}{6} = -0.8283$ - $\bar{X}_1 = \frac{-0.41 + 0.40 + (-0.86) + 1.69 + 0.46 + 1.40}{6} = \frac{2.68}{6} = 0.4467$ **Step 2: Calculate covariance and variance** Using the sample covariance formula with n-1 denominator: - $\text{Cov}(Y, X_1) = 0.5633$ - $\text{Var}(X_1) = 1.0000$ **Step 3: Calculate $\hat{\beta}_1$** $$\hat{\beta}_1 = \frac{0.5633}{1.0000} = 0.5633$$ **Step 4: Calculate $\hat{\alpha}$** $$\hat{\alpha} = \bar{Y} - \hat{\beta}_1 \bar{X}_1 = -0.8283 - (0.5633 \times 0.4467) = -0.8283 - 0.2517 = -1.080$$ Therefore, the estimated parameters are $\hat{\alpha} = -1.080$ and $\hat{\beta}_1 = 0.5633$, which matches option A.

Author: Tanishq Prabhu