Questions: Government agencies keep data about the income distribution of the population. The Ramirez family and Cooper family live in a county with 6000 families. The Ramirez family's income is at the 88th percentile. The Cooper family's income is at the 16th percentile. (a) Which of the following must be true about the Ramirez family's and the Cooper family's incomes? - The Ramirez family earns more than the Cooper family. - Both the Ramirez family and the Cooper family earn more than the median income. - The Ramirez family earns 72,000 more than the Cooper family. - The Ramirez family and the Cooper family both have incomes in the bottom half of incomes in their county. (b) Which of the following must be true about the Ramirez family's income? - The Ramirez family earns more than about 12% of families in their county. - About 12% of the families in their county earn more than the Ramirez family. - The Ramirez family earns about 88% of the highest income in their county. - The Ramirez family earns about 12% of the highest income in their county.

Solution Steps

1. The Ramirez family earns more than the Cooper family.

   • Since the Ramirez family's income is at the \(88^{\text{th}}\) percentile and the Cooper family's income is at the \(16^{\text{th}}\) percentile, it follows that: \[ \text{Income}_{\text{Ramirez}} > \text{Income}_{\text{Cooper}} \] Thus, this statement is True.

2. Both the Ramirez family and the Cooper family earn more than the median income.

   • The median income corresponds to the \(50^{\text{th}}\) percentile. The Cooper family, being at the \(16^{\text{th}}\) percentile, earns less than the median: \[ \text{Income}_{\text{Cooper}} < \text{Median} \] Therefore, this statement is False.

3. The Ramirez family earns \$72,000 more than the Cooper family.

   • Without specific income values, we cannot determine the exact difference in income. Thus, this statement is Cannot be determined.

4. The Ramirez family and the Cooper family both have incomes in the bottom half of incomes in their county.

   • The Ramirez family is at the \(88^{\text{th}}\) percentile, which is above the median, indicating they are in the top half: \[ \text{Income}_{\text{Ramirez}} > \text{Median} \] Hence, this statement is False.

1. The Ramirez family earns more than about \(12\%\) of families in their county.

   • Since the Ramirez family is at the \(88^{\text{th}}\) percentile, they earn more than \(88\%\) of families: \[ \text{Income}_{\text{Ramirez}} > 88\% \text{ of families} \] Thus, this statement is False.

2. About \(12\%\) of the families in their county earn more than the Ramirez family.

   • Since the Ramirez family is at the \(88^{\text{th}}\) percentile, it follows that: \[ 100\% - 88\% = 12\% \] Therefore, this statement is True.

3. The Ramirez family earns about \(88\%\) of the highest income in their county.

   • Percentiles do not directly translate to a percentage of the highest income. Thus, this statement is False.

4. The Ramirez family earns about \(12\%\) of the highest income in their county.

   • Similar to the previous point, this statement is also False.

Final Answer