STATISTICS Interpreting percentile ranks Government agencies keep data about the income distribution of the population. The Tran family and Baker family live in a county with 11,000 families. The Tran family's income is at the $91^{\text {st }}$ percentile. The Baker family's income is at the $34^{\text {th }}$ percentile. (a) Which of the following must be true about the Tran family's and the Baker family's incomes? Both the Tran family and the Baker family earn more than the median income. Both the Tran family and the Baker family earn less than the median income. The Tran family earns more than the Baker family. The Baker family earns more than the Tran family. (b) Which of the following must be true about the Tran family's income? About $91 \%$ of the families in their county earn less than the Tran family. The Tran family earns about $91 \%$ of the highest income in their county. The Tran family's income is in the bottom half of incomes in their county. The Tran family earns $\$ 91,000$.
Solution
Step 1 ：The percentile rank of a score is the percentage of scores in its frequency distribution that are equal to or lower than it. In this case, if the Tran family's income is at the 91st percentile, it means that 91% of the families in the county earn less than or equal to the Tran family. Similarly, if the Baker family's income is at the 34th percentile, it means that 34% of the families in the county earn less than or equal to the Baker family.
Step 2 ：For part (a), since the Tran family's income is at the 91st percentile and the Baker family's income is at the 34th percentile, it is clear that the Tran family earns more than the Baker family.
Step 3 ：For part (b), since the Tran family's income is at the 91st percentile, it means that about 91% of the families in their county earn less than the Tran family.
Step 4 ：Final Answer: \(\boxed{\text{(a) The Tran family earns more than the Baker family.}}\)
Step 5 ：Final Answer: \(\boxed{\text{(b) About 91\% of the families in their county earn less than the Tran family.}}\)
