Answer: α̂ = -1.080, β̂₁ = 0.5633

## Explanation The correct answer is **A** (α̂ = -1.080, β̂₁ = 0.5633). ### Step-by-Step Calculation: 1. **Calculate β̂₁**: β̂₁ = Cov(Y, X₁) / Var(X₁) From the table: - Cov(Y, X₁) = (1/(n-1)) Σ(Y - Ȳ)(X₁ - X̄₁) = (1/5) * 2.759533 = 0.5519066 - Var(X₁) = 0.9797 (given in the explanation) β̂₁ = 0.5519066 / 0.9797 = 0.5633 2. **Calculate α̂**: α̂ = Ȳ - β̂₁ * X̄₁ From the table: - Ȳ = -0.82833 - X̄₁ = 0.446667 α̂ = -0.82833 - (0.5633 * 0.446667) = -0.82833 - 0.2516 = -1.07993 ≈ -1.080 ### Verification: The calculations match option A exactly: - α̂ = -1.080 - β̂₁ = 0.5633 This is a standard linear regression estimation problem using the ordinary least squares (OLS) method. The intercept (α̂) is calculated using the formula α̂ = Ȳ - β̂₁X̄₁, and the slope coefficient (β̂₁) is calculated using β̂₁ = Cov(Y, X₁)/Var(X₁).

Author: Nikitesh Somanthe