Answer: $367

## Explanation To compute the bonus, we apply the following linear transformation to each employee's salary: $$Y = \alpha + \beta X$$ $$Y = 1,000 + 0.15X$$ where: - Y is the transformed variable (bonus) - X is the original variable (salary) - β = 0.15 (scale constant) - α = 1,000 (shift constant) The variance of Y (bonuses) is given by: $$Var(Y) = Var(\alpha + \beta X) = \beta^2 Var(X) = \beta^2 \sigma^2$$ Given: - Variance of salary $Var(X) = 6,000,000$ - β = 0.15 Therefore: $$Var(Y) = 0.15^2 \times 6,000,000 = 0.0225 \times 6,000,000 = 135,000$$ Standard deviation is the square root of variance: $$SD(Y) = \sqrt{135,000} = \$367.42 \approx \$367$$ **Key Insight**: When applying a linear transformation Y = α + βX: - The shift constant α affects the mean but not the variance - The scale constant β affects both mean and variance - Standard deviation scales by |β|, while variance scales by β²

Author: Tanishq Prabhu