Question

government agencies keep data about the income distribution of the population. the allen family and stewart family live in a county with 9000 families. the allen familys income is at the 32nd percentile. the stewart familys income is at the 66th percentile. (a) which of the following must be true about the allen familys income? ○ the allen family earns more than about 68% of families in their county. ○ about 68% of the families in their county earn more than the allen family. ○ the allen family earns about 32% of the highest income in their county. ○ the allen family earns about 68% of the highest income in their county. (b) which of the following must be true about the allen familys and the stewart familys incomes? ○ the stewart family earns more than the allen family. ○ both the allen family and the stewart family earn more than the median income. ○ the stewart family earns $34,000 more than the allen family. ○ the allen family and the stewart family both have incomes in the bottom half of incomes in their county.

Explanation:

Step1: Understanding Percentile Concept

A percentile \( p \) means that \( p\% \) of the data values are less than or equal to that value. If the Allen family's income is at the \( 32^{nd} \) percentile, about \( 32\% \) of families have income less than or equal to the Allen family's income, and about \( 100 - 32=68\% \) of families have income more than the Allen family's income.

Step2: Analyzing Option for (a)

• Option "The Allen family earns more than about \( 68\% \) of families in their county": Incorrect, because if at \( 32^{nd} \) percentile, \( 32\% \) of families have income \( \leq \) Allen's income, so \( 68\% \) have income \(>\) Allen's income.
• Option "About \( 68\% \) of the families in their county earn more than the Allen family": Correct, as per percentile definition (\( 100 - 32 = 68\)).
• Option "The Allen family earns about \( 32\% \) of the highest income in their county": Incorrect, percentile is about the position in the data distribution, not a proportion of the maximum value.
• Option "The Allen family earns about \( 68\% \) of the highest income in their county": Incorrect, same reason as above.

Step3: Using Relationship between Percentiles for (b)

If Allen is at \( 32^{nd} \) percentile and Stewart is at \( 66^{th} \) percentile.

• Option "The Stewart family earns more than the Allen family": Since \( 66>32 \), the Stewart family's income is higher than the Allen family's income.
• Option "Both the Allen family and the Stewart family earn more than the median income": Median is \( 50^{th} \) percentile. Allen (\( 32^{nd} \)) is less than median (\( 50^{th} \)), so this option is incorrect.
• Option "The Stewart family earns \( \$34000 \) more than the Allen family": There is no information about the actual income difference in terms of dollars from the percentile data.
• Option "The Allen family and the Stewart family both have incomes in the bottom half of incomes in their county": Bottom - half is \( 0 - 50^{th} \) percentile. Stewart (\( 66^{th} \)) is in the top - half (\( 50 - 100^{th} \)), so this option is incorrect.

Answer:

(a) About \( 68\% \) of the families in their county earn more than the Allen family.
(b) The Stewart family earns more than the Allen family.