Explanation

To calculate the adjusted R², we first need to find the regular R² and then apply the adjustment formula.

Step 1: Calculate R²

$R^2 = 1 - \frac{SSR}{TSS} = 1 - \frac{100}{500} = 1 - 0.2 = 0.8$

Where:

• SSR (Sum of Squared Residuals) = 100
• TSS (Total Sum of Squares) = 500

Step 2: Calculate Adjusted R²

$\bar{R}^2 = 1 - \left(\frac{n - 1}{n - k - 1}\right)(1 - R^2)$

Where:

• n = number of observations = 50
• k = number of independent variables = 4
• R² = 0.8

$\bar{R}^2 = 1 - \left(\frac{50 - 1}{50 - 4 - 1}\right)(1 - 0.8) = 1 - \left(\frac{49}{45}\right)(0.2) = 1 - (1.0889 \times 0.2) = 1 - 0.2178 = 0.7822 \approx 0.78$

Why Adjusted R² is Lower

• The adjusted R² penalizes for adding more independent variables
• With 4 predictors and only 50 observations, there's a penalty for model complexity
• The regular R² of 0.8 is adjusted downward to 0.78 to account for the number of predictors

Therefore, the correct answer is B: 0.78