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).