Question

government agencies keep data about the income distribution of the population. the diaz family and kim family live in a county with 9000 families. the diaz familys income is at the 64th percentile. the kim familys income is at the 27th percentile.
(a) which of the following must be true about the diaz familys income?
about 64% of the families in their county earn less than the diaz family.
the diaz family earns about 64% of the highest income in their county.
the diaz familys income is in the bottom half of incomes in their county.
the diaz family earns $64,000.
(b) which of the following must be true about the diaz familys and the kim familys incomes?
both the diaz family and the kim family earn more than the median income.
both the diaz family and the kim family earn less than the median income.
the diaz family earns more than the kim family.
the kim family earns more than the diaz family.

Explanation:

Step1: Understand percentile concept

The \(n^{th}\) percentile means that \(n\%\) of the data values are less than that value.

Step2: Analyze part (a)

• For the Diaz family at \(64^{th}\) percentile: By definition of percentile, about \(64\%\) of the families in their county earn less than the Diaz family.
• The Diaz family's income is not necessarily \(64\%\) of the highest income (no such relation from percentile definition).
• Since \(64\%>50\%\), the Diaz family's income is not in the bottom - half.
• There is no information to suggest the Diaz family earns \(\$64000\).

Step3: Analyze part (b)

• Median is the \(50^{th}\) percentile. The Diaz family (\(64^{th}\) percentile) earns more than the Kim family (\(27^{th}\) percentile).
• The Diaz family (\(64^{th}>50^{th}\)) earns more than the median, while the Kim family (\(27^{th}<50^{th}\)) earns less than the median.

Answer:

(a) About \(64\%\) of the families in their county earn less than the Diaz family.
(b) The Diaz family earns more than the Kim family.