Answer: $112

## Explanation The compensation structure is given by: \[ Y = \alpha + \beta X \] where: - \( Y \) = total compensation - \( \alpha = 30,000 \) = base salary - \( \beta = 0.05 \) = bonus rate per dollar of sales - \( X \) = sales amount Given: - Mean sales = $300,000 per year - Variance of sales, \( \text{Var}(X) = 5,000,000 \) **Step 1: Variance of compensation** For a linear transformation \( Y = \alpha + \beta X \): \[ \text{Var}(Y) = \beta^2 \cdot \text{Var}(X) \] Substitute the values: \[ \text{Var}(Y) = (0.05)^2 \times 5,000,000 \] \[ \text{Var}(Y) = 0.0025 \times 5,000,000 \] \[ \text{Var}(Y) = 12,500 \] **Step 2: Standard deviation of compensation** Standard deviation is the square root of variance: \[ \sigma_Y = \sqrt{\text{Var}(Y)} = \sqrt{12,500} \] \[ \sigma_Y \approx 111.8 \] Rounded to the nearest dollar: **$112** **Key points:** - The base salary (\( \alpha \)) does not affect the variance because it's a constant shift - Only the bonus component (\( \beta X \)) contributes to the variance - The variance scales by \( \beta^2 \) when multiplying a random variable by a constant Thus, the correct answer is **D) $112**.

Author: Nikitesh Somanthe