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.