Chapter 5, Problem 8P

As was discussed in the Analytics in Action from Chapter 2, the U.S. Census Bureau is a leading source of quantitative data related to the people and economy of the United States. The crosstabulation below represents the number of households (1,000s) and the household income by the highest level of education for the head of household (U.S. Census Bureau web site, 2013). Use this crosstabulation to answer the following questions.

1. a. Develop a joint probability table.
2. b. What is the probability the head of one of these households has a master’s degree or higher education?
3. c. What is the probability a household is headed by someone with a high school diploma earning $100,000 or more?
4. d. What is the probability one of these households has an income below $25,000?
5. e. What is the probability a household is headed by someone with a bachelor’s degree earning less than $25,000?
6. f. Are household income and educational level independent?

To determine

Create a joint probability table.

Answer to Problem 8P

The joint probability table is given below:

Highest Level of Education	Household income
	Under $25,000	$25,000 to $49,999	$49,999 to $50,000	$100,000 and Over
High school graduate	0.1505	0.1519	0.1438	0.0530
Bachelor’s degree	0.0378	0.0634	0.1168	0.1191
Master’s degree	0.0104	0.0184	0.0460	0.0624
Doctoral degree	0.0012	0.0024	0.0064	0.0164

Explanation of Solution

The joint probability is calculated by dividing the intersection value of two events by the overall total.

The joint probability table is given below:

Highest Level of Education	Household income				Total
	Under $25,000	$25,000 to $49,999	$49,999 to $50,000	$100,000 and Over
High school graduate	$\begin{array}{r}\frac{9,880}{65,644}\\ =0.1505\end{array}$	$\begin{array}{r}\frac{9,970}{65,644}\\ =0.1519\end{array}$	$\begin{array}{r}\frac{9,441}{65,644}\\ =0.1438\end{array}$	$\begin{array}{r}\frac{3,482}{65,644}\\ =0.0530\end{array}$	0.4993
Bachelor’s degree	$\begin{array}{r}\frac{2,484}{65,644}\\ =0.0378\end{array}$	$\begin{array}{r}\frac{4,164}{65,644}\\ =0.0634\end{array}$	$\begin{array}{r}\frac{7,666}{65,644}\\ =0.1168\end{array}$	$\begin{array}{r}\frac{7,817}{65,644}\\ =0.1191\end{array}$	0.3371
Master’s degree	$\begin{array}{r}\frac{685}{65,644}\\ =0.0104\end{array}$	$\begin{array}{r}\frac{1,205}{65,644}\\ =0.0184\end{array}$	$\begin{array}{r}\frac{3,019}{65,644}\\ =0.0460\end{array}$	$\begin{array}{r}\frac{4,094}{65,644}\\ =0.0624\end{array}$	0.1371
Doctoral degree	$\begin{array}{r}\frac{79}{65,644}\\ =0.0012\end{array}$	$\begin{array}{r}\frac{160}{65,644}\\ =0.0024\end{array}$	$\begin{array}{r}\frac{422}{65,644}\\ =0.0064\end{array}$	$\begin{array}{r}\frac{1,076}{65,644}\\ =0.0164\end{array}$	0.0265
Total	0.2000	0.2361	0.3130	0.2509	1

To determine

Find the probability of the head of one of these households who has a Master’s degree or higher education.

Answer to Problem 8P

The probability the head of one of these households has a Master’s degree or higher education is 0.1636.

Explanation of Solution

The probability that the head of one of these households has a Master’s degree or higher education is calculated as follows:

$\begin{array}{c}P\left(\text{Master's degree or higher education}\right)=P\left(\text{Master's degree}\right)+P\left(\text{Doctoral degree}\right)\\ =\frac{9,003}{65,644}+\frac{1,737}{65,644}\\ =0.1371+0.0265\\ =0.1636\end{array}$

Therefore, the probability that the head of one of these households has a Master’s degree or higher education is 0.1636.

To determine

Find the probability that a household is headed by someone with a high school diploma earning $100,000 or more.

Answer to Problem 8P

The probability that a household is headed by someone with a high school diploma earning $100,000 or more is 0.1062.

Explanation of Solution

The probability that a household is headed by someone with a high school diploma earning $100,000 or more is calculated as follows:

$\begin{array}{c}P\left(\text{\$100,000 or more}|\text{high school diploma}\right)=\frac{P\left(\text{\$100,000 or more}\cap \text{high school diploma}\right)}{P\left(\text{high school diploma}\right)}\\ =\frac{\left(\frac{3,482}{65,644}\right)}{\left(\frac{32,773}{65,644}\right)}\\ =0.1062\end{array}$

Therefore, the probability that a household is headed by someone with a high school diploma earning $100,000 or more is 0.1062.

To determine

Find the probability that one of the households has an income below $25,000.

Answer to Problem 8P

The probability that one of the households has an income below $25,000 is 0.2.

Explanation of Solution

The probability that one of the households has an income below $25,000 is calculated as follows:

$\begin{array}{c}P\left(\text{income below \$25,000}\right)=\frac{13,128}{65,644}\\ =0.2\end{array}$

The probability that one of the households has an income below $25,000 is 0.2.

To determine

Find the probability that a household is headed by someone with a Bachelor’s degree earning less than $25,000.

Answer to Problem 8P

The probability that a household is headed by someone with a Bachelor’s degree earning less than $25,000 is 0.1122.

Explanation of Solution

The probability that a household is headed by someone with a Bachelor’s degree earning less than $25,000 is calculated as follows:

$\begin{array}{c}P\left(\text{under \$25,000|Bachelor's degree}\right)=\frac{P\left(\text{under \$25,000}\cap \text{Bachlor's degree}\right)}{P\left(\text{Bachlor's degree}\right)}\\ =\frac{\left(\frac{2,484}{65,644}\right)}{\left(\frac{22,131}{65,644}\right)}\\ =0.1122\end{array}$

The probability that a household is headed by someone with a Bachelor’s degree earning less than $25,000 is 0.1122.

To determine

Check whether the household income and educational level are independent.

Answer to Problem 8P

The household income and educational level are not independent.

Explanation of Solution

Two events are said to be independent if $P\left(A|B\right)$ is equal to $P\left(A\right)$.

From Part (c), it is obtained that $P\left(\text{\$100,000 or more}|\text{high school diploma}\right)=0.1062$.

The probability of $100,000 or more is calculated as follows:

$\begin{array}{c}P\left(\text{\$100,000 or more}\right)=\frac{16,469}{65,644}\\ =0.2509\end{array}$

It is clear that $P\left(\text{\$100,000 or more}|\text{high school diploma}\right)\ne P\left(\text{\$100,000 or more}\right)$. Therefore, the household income and educational level are not independent.