Question

government agencies keep data about the income distribution of the population. the harris family and clark family live in a county with 5000 families. the harris familys income is at the 58th percentile. the clark familys income is at the 69th percentile. (if necessary, consult a list of formulas.) (a) which of the following must be true about the harris familys and the clark familys incomes? the harris family earns more than the clark family. both the harris family and the clark family earn more than the median income. the clark family earns $11,000 more than the harris family. the harris family and the clark family both have incomes in the bottom half of incomes in their county. (b) which of the following must be true about the harris familys income? the harris family earns less than about 58% of families in their county. the harris family earns about 58% of the highest income in their county. the harris family earns more than about 58% of families in their county. the harris family earns about 42% of the highest income in their county.

Explanation:

For part (a):

• The \(n^{th}\) percentile means that \(n\%\) of the data is less than or equal to that value. Since \(58^{th}\) and \(69^{th}\) percentiles are both above the \(50^{th}\) percentile (median), both families earn more than the median income.
• The Harris family (\(58^{th}\) percentile) does not earn more than the Clark family (\(69^{th}\) percentile). We don't know the exact income difference (\(11000\) is arbitrary). And they are not in the bottom - half (bottom - half is below \(50^{th}\) percentile).

For part (b):

• By the definition of percentile, the \(n^{th}\) percentile means that the value is greater than \(n\%\) of the data. So the Harris family (\(58^{th}\) percentile) earns more than about \(58\%\) of families in their county.

Answer:

(a) Both the Harris family and the Clark family earn more than the median income.
(b) The Harris family earns more than about \(58\%\) of families in their county.