Chapter 6.2, Problem 57E

To determine

Interpretation and calculation of the standard deviation of the sum S of two independent variable X and Y .

Answer to Problem 57E

Standard deviation of the sum S ,

  ${\sigma }_S\approx \$163.13$

Explanation of Solution

Given information:

X and Y are independent random variables.

For X :

Mean,

  ${\mu }_X=732.50$

Standard deviation,

  ${\sigma }_X=103$

For Y :

Mean,

  ${\mu }_Y=825$

Standard deviation,

  ${\sigma }_Y=126.50$

Calculations:

We know that

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

Now,

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

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

When the random variables are independent, the variance of the sum is equal to the sum of their variances.

  ${\sigma }_{X+Y}^2={\sigma }_X^2+{\sigma }_Y^2={\left(103\right)}^2+{\left(126.50\right)}^2=\$26611.25$

We also know that

The standard deviation is the square root of the variance:

  ${\sigma }_{X+Y}=\sqrt{{\sigma }_{X+Y}^2}=\sqrt{26611.25}=\$163.13$

On an average, the total amount spent on both the Downtown campus and the Main campus deviates by $163.3 from the mean amount spent of $1,557.50.