Chapter 5.1, Problem 22E

(a)

To determine

To find:

The expected value for the cost of books for the semester.

Answer to Problem 22E

The expected value of the cost of books for Liberal Arts Major is 516.97

Explanation of Solution

Given:

In the following probability distribution for the cost of textbooks in a fall semester for various liberal arts majors, each probability represents the chance that the total cost of a student’s books will be the given amount.

Cost of Textbooks for Liberal Arts Majors
X	P(X = x)
$262	0.19
$410	0.21
$590	0.17
$653	0.43

Formula Used:

The expected value for a discrete random variable X is equal to the mean of the probability distribution of X and is given by

$E(X)={\displaystyle \sum \left[x_i.P\left(X=x_i\right)\right]}$

Where $x_i$ is the $i^{th}$ value of the random variable X.

Calculation:

The expected value of the cost of books for Liberal Arts Major will be:

$\begin{array}{rll}E(X)& =& {\displaystyle \sum \left[x_i.P\left(X=x_i\right)\right]}\\ & =& \left(262\times 0.19+410\times 0.21+590\times 0.17+653\times 0.43\right)\\ & =& 49.78+86.1+100.3+280.79\\ & =& 516.97\end{array}$

Therefore, the expected value of the cost of books for Liberal Arts Major is 516.97.

(b)

To determine

To find:

The expected value for the cost of books for the semester.

Answer to Problem 22E

The expected value of the cost of books for Business Majors is 389.02

Explanation of Solution

Given:

In the following probability distribution for the cost of textbooks in a fall semester for various business majors, each probability represents the chance that the total cost of a student’s books will be the given amount.

Cost of Textbooks for Liberal Arts Majors
X	P(X = x)
$378	0.35
$389	0.14
$392	0.25
$401	0.26

Formula Used:

The expected value for a discrete random variable X is equal to the mean of the probability distribution of X and is given by

$E(X)={\displaystyle \sum \left[x_i.P\left(X=x_i\right)\right]}$

Where $x_i$ is the $i^{th}$ value of the random variable X.

Calculation:

The expected value of the cost of books for Liberal Arts Major will be:

$\begin{array}{rll}E(X)& =& {\displaystyle \sum \left[x_i.P\left(X=x_i\right)\right]}\\ & =& \left(378\times 0.35+389\times 0.14+392\times 0.25+401\times 0.26\right)\\ & =& 132.3+54.46+98+104.26\\ & =& 389.02\end{array}$

Therefore, the expected value of the cost of books for Business Majors is 389.02.

(c)

To determine

To find:

The group of majors which is more likely to feel that their textbooks are priced fairly.

Answer to Problem 22E

The group of majors which is more likely to feel that their textbooks are priced fairly is Business Majors.

Explanation of Solution

Formula Used:

The variance for a discrete probability distribution of random variable X is given by

$\begin{array}{rll}{\sigma }^2& =& {\displaystyle \sum \left[x_i{}^2.P\left(X=x_i\right)\right]}-{\mu }^2\\ & =& {\displaystyle \sum \left[{\left(x_i-\mu \right)}^2.P(X=x_i)\right]}\end{array}$

Where $x_i$ is the $i^{th}$ value of the random variable X and $\mu$ is the mean of the probability distributions.

Calculation:

The standard deviation of Liberal Arts Major is:

$\begin{array}{rll}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{{\displaystyle \sum \left[x_i{}^2.P(X=x_i)\right]}-{\mu }^2}\\ & =& \sqrt{\left[{262}^2\times 0.19+{410}^2\times 0.21+{590}^2\times 0.17+{653}^2\times 0.43\right]-{516.97}^2}\\ & =& 153.68\end{array}$

The standard deviation of Business major is:

$\begin{array}{rll}\sigma & =& \sqrt{{\sigma }^2}\\ & =& \sqrt{{\displaystyle \sum \left[x_i{}^2.P(X=x_i)\right]}-{\mu }^2}\\ & =& \sqrt{\left[{378}^2\times 0.35+{389}^2\times 0.14+{392}^2\times 0.25+{401}^2\times 0.26\right]-{389.02}^2}\\ & =& 9.07\end{array}$

Comparing the standard deviation, the variation in prices is lesser in Business Major Books than the variation in prices in Liberal Arts Major.

Therefore, Business Major books is more likely to feel that their textbooks are priced fairly.