Answer: $112

## Explanation To compute the standard deviation of compensation, we apply linear transformation principles to sales data: ### Step 1: Define the Compensation Formula Compensation (Y) is given by: ``` Y = α + βX Y = 30,000 + 0.05X ``` where: - Y = compensation (transformed variable) - X = sales variable - β = scale constant ($0.05/$1) - α = shift constant ($30,000) ### Step 2: Calculate Variance of Compensation For linear transformations, the variance of Y is: ``` Var(Y) = Var(α + βX) = β²Var(X) = β²σ² ``` Given: - Var(X) = σ² = 5,000,000 - β = 0.05 ``` Var(Y) = (0.05)² × 5,000,000 = 0.0025 × 5,000,000 = 12,500 ``` ### Step 3: Calculate Standard Deviation Standard deviation is the square root of variance: ``` SD(Y) = √Var(Y) = √12,500 = $111.8 ≈ $112 ``` **Key Insight:** The basic salary ($30,000) doesn't affect the variance or standard deviation since it's a constant shift that gets canceled out in variance calculations. Only the variable component (bonus) contributes to the dispersion of compensation.

Author: Tanishq Prabhu