To determine Tiffany's conclusions regarding income and education, we start by calculating relevant probabilities from the given data. The table contains information about the income distribution among people with different levels of education:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
& <\$40,000 & \$40,000-\$60,000 & >\$60,000 & \text{Total} \\
\hline
\text{High School Diploma} & 51 & 19 & 6 & 76 \\
\hline
\text{Bachelor's Degree} & 24 & 40 & 17 & 81 \\
\hline
\text{Master's Degree} & 3 & 22 & 48 & 73 \\
\hline
\text{Total} & 78 & 81 & 71 & 230 \\
\hline
\end{array}
\][/tex]
### Calculation of Probabilities
1. Given a person has only a high school diploma, calculate the probabilities of different income levels:
[tex]\[
P(\text{Income} < \$40,000 \mid \text{High School Diploma}) = \frac{51}{76} \approx 0.671
\][/tex]
[tex]\[
P(\$40,000 \leq \text{Income} \leq \$60,000 \mid \text{High School Diploma}) = \frac{19}{76} \approx 0.25
\][/tex]
[tex]\[
P(\text{Income} > \$60,000 \mid \text{High School Diploma}) = \frac{6}{76} \approx 0.079
\][/tex]
Based on these probabilities, Tiffany can conclude that a person with only a high school diploma is most likely to have an income less than \[tex]$40,000, given the highest probability is \(0.671\).
2. Given a person has an income greater than \$[/tex]60,000, calculate the probabilities of different education levels:
[tex]\[
P(\text{High School Diploma} \mid \text{Income} > \$60,000) = \frac{6}{71} \approx 0.085
\][/tex]
[tex]\[
P(\text{Bachelor's Degree} \mid \text{Income} > \$60,000) = \frac{17}{71} \approx 0.239
\][/tex]
[tex]\[
P(\text{Master's Degree} \mid \text{Income} > \$60,000) = \frac{48}{71} \approx 0.676
\][/tex]
Based on these probabilities, Tiffany can conclude that a person with an income greater than \[tex]$60,000 is most likely to have a master's degree, given the highest probability is \(0.676\).
### Conclusions
1. Given a person has only a high school diploma, they are most likely to have an income less than $[/tex]40,000:
This conclusion is supported by the calculated probability [tex]\( P(\text{Income} < \$40,000 \mid \text{High School Diploma}) \approx 0.671 \)[/tex], which is the highest among the three income ranges.
2. Given a person has an income greater than [tex]$60,000, it is most likely that their highest level of education is a master's degree:
This conclusion is supported by the calculated probability \( P(\text{Master's Degree} \mid \text{Income} > \$[/tex]60,000) \approx 0.676 \), which is the highest among the three education levels.
In summary, Tiffany can conclude that low income is most common among those with only a high school diploma, and high income is most common among those with a master's degree.
