Janet is planning to rent a booth at a festival for a day to sell clothes that she has made.She sells jackets for $\$ 80$ and skirts for $\$ 50$ Her past experiences suggests that sales of jackets will have a mean of 6.5 with a standard deviation of $\sigma _ { j }$ ,and sales of skirts will have a mean of 12.6 with a standard deviation of $\sigma _ { s }$ .The cost of renting the booth for the day is $\$ 180$ Let the random variable J represent the number of jackets that Janet sells and the random variable S represent the number of skirts that Janet sells.Then Janet's net income,I,is given by the expression: I = 80J + 50S - 180
Find an expression for the standard deviation of Janet's net income.Assume that sales are independent of each other.
A)? = 80 $\sigma _ { j }$ + 50 $\sigma _ { s }$
B)? = $\sqrt { 6400 \sigma _ { j } ^ { 2 } + 2500 \sigma _ { s } ^ { 2 } - 180 }$
C)? = $\sqrt { 80 \sigma _ { j } ^ { 2 } + 50 \sigma _ { s } ^ { 2 } }$
D)? = $\sqrt { 6400 \sigma _ { j } ^ { 2 } + 2500 \sigma _ { s } ^ { 2 } }$
E)? = 80 $\sigma _ { j }$ + 50 $\sigma _ { s }$ - 180

Question

Quizplus · Accepted Answer

The standard deviation of Janet's net income, given that sales of jackets and skirts are independent, can be found by taking the square root of the sum of the variances of each component of her income, multiplied by the square of their respective coefficients. The variance of income from jackets is $6400\sigma_j^2$ (since variance is squared, $80^2\sigma_j^2$), and from skirts is $2500\sigma_s^2$ ($50^2\sigma_s^2$). The cost of renting the booth does not affect the standard deviation as it is a constant. Therefore, the correct expression is $\sqrt{6400\sigma_j^2 + 2500\sigma_s^2}$.

Janet Is Planning to Rent a Booth at a Festival $80\$ 80$80

Janet Is Planning to Rent a Booth at a Festival $\$ 80$