Question

Community college costs : Refer to Exercise 49.

Note that $X\mathrm{and}Y$ are independent random variables because the two students are randomly selected from each of the campuses. Calculate and interpret the standard deviation of the sum $S=X+Y$.

Short answer

The total amount spent on both the Downtown and Main campuses differs by $\mathbf{\$}\mathbf{163}\mathbf{.}\mathbf{3}$ on average from the mean amount spent of$\$1,557.50$.

Step-by-step solution

Given Information

Given :

$X$: The amount spent on tuition by a randomly selected student at Main campus

$Y$: The amount spent on tuition by a randomly selected student at Downtown campus

Calculating and interpreting the standard deviation of the sumS = X + Y.

Total mean of the mean amount spent by both X and Y is given by :

${\mu }_{X+Y}=732.50+825=\$1,557.50$

The variance of the total is equal to the sum of the variances of the random variables when they are independent.

role="math" localid="1654176112869" $$\begin{gathered} {{\sigma }^2}_{X+Y}={{\sigma }^2}_X+{{\sigma }^2}_Y \\ ={\left(103\right)}^2+{(126.50)}^2 \\ =\$26611.25 \end{gathered}$$

We also know that the standard deviation equals the variance squared:

$$\begin{gathered} {\sigma }_{X+Y}=\sqrt{{{\sigma }^2}_{X+Y}} \\ =\sqrt{26611.25} \\ =\mathbf{\$}\mathbf{163}\mathbf{.}\mathbf{13} \end{gathered}$$