Question

government agencies keep data about the income distribution of the population. the diaz family and johnson family live in a county with 6000 families. the diaz familys income is at the 29th percentile. the johnson familys income is at the 72nd percentile. (if necessary, consult a list of formulas.) (a) which of the following must be true about the diaz familys income? about 29% of the families in their county earn less than the diaz family. the diaz family earns about 29% of the highest income in their county. the diaz familys income is in the top half of incomes in their county. the diaz family earns $29,000. (b) which of the following must be true about the diaz familys and the johnson familys incomes? both the diaz family and the johnson family earn more than the median income. both the diaz family and the johnson family earn less than the median income. the johnson family earns more than the diaz family. the diaz family earns more than the johnson family.

Explanation:

• Part (a):
• The definition of a percentile: If a value is at the \(p^{th}\) percentile, approximately \(p\%\) of the data values are less than that value. So, if the Diaz family is at the \(29^{th}\) percentile, about \(29\%\) of the families in their county earn less than the Diaz family.
• The \(29^{th}\) percentile has nothing to do with the percentage of the highest - income (eliminating the second option). The top half of incomes would start at the \(50^{th}\) percentile, and \(29<50\) (eliminating the third option). There is no information about the actual income amount (eliminating the fourth option).
• Part (b):
• The median is the \(50^{th}\) percentile. Since \(29<50\) (Diaz is at \(29^{th}\) percentile) and \(72 > 50\) (Johnson is at \(72^{nd}\) percentile), the first two options (both above or both below the median) are wrong. Also, since \(29<72\), the Johnson family (at \(72^{nd}\) percentile) earns more than the Diaz family (at \(29^{th}\) percentile), and the fourth option (Diaz earns more than Johnson) is wrong.

Answer:

(a) About \(29\%\) of the families in their county earn less than the Diaz family.
(b) The Johnson family earns more than the Diaz family.