Question

A salesperson receives an annual salary of $6,000 plus 8% of the value of the orders she takes. The annual value of these orders can be represented by a random variable with a mean of $600,000 and a standard deviation of $180,000. Find the mean and standard deviation of the salesperson’s annual income.

Answer 1

• Mean: $48,000

• Standard deviation: $14,400

Step-by-step explanation:

1. Mean of the annual income:

The mean income is the expected income, which is: the sum of the annual salary (constant) plus the 8% of the mean value of the orders ($600,000):

• Mean annual income = $6,000 + 8% × $600,000 = $6,000 + $48,000 = $54,000.

2. Standard deviation of the annual income.

The standar deviation is a measure of how extended the values are.

It means that the annual value of the orders will be around the mean plus or minus a number of standard deviations, depending on the precision you want.

The 8% of the the standard deviation is 8% × $180,000 = $14,400.

Since the $6,000 is a constant it does not modify the standard deviation.

These results are a consequence of the linearity of the mean and the standard deviation.

Call Y the salesperson salary, and X the valueof the orders. Then:

• Y = 6,000 + 8% X

The linearity property states that:

• Mean of Y = 8% × (mean of X) + 6,000

And:

• Standard deviation of Y = 8% × (Standard deviation of X).