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

## Explanation The correct answer is B: $\hat{Y} = -1.0799 + 0.5633 X_1$ To calculate the estimated regression equation $\hat{Y} = \hat{\alpha} + \hat{\beta}_1 X_1$, we need to compute: 1. **$\hat{\beta}_1$**: The slope coefficient 2. **$\hat{\alpha}$**: The intercept ### Step 1: Calculate means From the data: - Y values: -2, -0.11, -1.68, -0.36, -0.08, -0.74 - X₁ values: -0.41, 0.40, -0.86, 1.69, 0.46, 1.40 Mean of Y = (-2 - 0.11 - 1.68 - 0.36 - 0.08 - 0.74)/6 = -5.97/6 = -0.995 Mean of X₁ = (-0.41 + 0.40 - 0.86 + 1.69 + 0.46 + 1.40)/6 = 2.68/6 = 0.4467 ### Step 2: Calculate Cov(Y, X₁) and Var(X₁) **Cov(Y, X₁)**: First compute deviations: - For each observation: (Yᵢ - Ȳ)(X₁ᵢ - X̄₁) - Sum these products and divide by (n-1) = 5 **Var(X₁)**: - Sum of squared deviations of X₁ from its mean, divided by (n-1) ### Step 3: Calculate $\hat{\beta}_1$ $\hat{\beta}_1 = \frac{\text{Cov}(Y, X_1)}{\text{Var}(X_1)}$ ### Step 4: Calculate $\hat{\alpha}$ $\hat{\alpha} = Ȳ - \hat{\beta}_1 X̄₁$ When performing these calculations with the given data: - $\hat{\beta}_1 ≈ 0.5633$ - $\hat{\alpha} ≈ -1.0799$ Thus, the estimated regression equation is: $\hat{Y} = -1.0799 + 0.5633 X_1$ This matches option B.

Author: Nikitesh Somanthe