Question

government agencies keep data about the income distribution of the population. the nelson family and miller family live in a county with 14,000 families. the nelson family’s income is at the 84th percentile. the miller family’s income is at the 18th percentile.
(if necessary, consult a list of formulas.)
(a) which of the following must be true about the nelson family’s income?
∘ about 84% of the families in their county earn less than the nelson family.
∘ the nelson family earns about 84% of the highest income in their county.
∘ the nelson family’s income is in the bottom half of incomes in their county.
∘ the nelson family earns $84,000.
(b) which of the following must be true about the nelson family’s and the miller family’s incomes?
∘ both the nelson family and the miller family earn more than the median income.
∘ both the nelson family and the miller family earn less than the median income.
∘ the nelson family earns more than the miller family.
∘ the miller family earns more than the nelson family.

Explanation:

Part (a)

Step 1: Recall the definition of a percentile

A percentile (e.g., the \( p^{\text{th}} \) percentile) of a data set is a value such that \( p\% \) of the data points are less than or equal to that value. For the Nelson family's income at the \( 84^{\text{th}} \) percentile, this means about \( 84\% \) of the families in the county have an income less than the Nelson family's income.

Step 2: Analyze each option

• Option 1: "About 84% of the families in their county earn less than the Nelson family." This matches the definition of the \( 84^{\text{th}} \) percentile.
• Option 2: "The Nelson family earns about 84% of the highest income in their county." A percentile does not relate to a percentage of the highest income, so this is incorrect.
• Option 3: "The Nelson family’s income is in the bottom half of incomes in their county." The \( 84^{\text{th}} \) percentile is above the \( 50^{\text{th}} \) percentile (median), so it's in the top half, not the bottom half. This is incorrect.
• Option 4: "The Nelson family earns $84,000." There's no information given to suggest the income is $84,000. This is incorrect.

Step 1: Recall the median (50th percentile)

The median is the \( 50^{\text{th}} \) percentile, meaning 50% of the data is below it and 50% is above it.

Step 2: Analyze the percentiles of the two families

• Nelson family: \( 84^{\text{th}} \) percentile (above the median, since \( 84 > 50 \)).
• Miller family: \( 18^{\text{th}} \) percentile (below the median, since \( 18 < 50 \)).

Step 3: Analyze each option

• Option 1: "Both the Nelson family and the Miller family earn more than the median income." Miller is at \( 18^{\text{th}} \) percentile (below median), so this is incorrect.
• Option 2: "Both the Nelson family and the Miller family earn less than the median income." Nelson is at \( 84^{\text{th}} \) percentile (above median), so this is incorrect.
• Option 3: "The Nelson family earns more than the Miller family." Since the \( 84^{\text{th}} \) percentile is higher than the \( 18^{\text{th}} \) percentile (a higher percentile means a higher income in the distribution), the Nelson family earns more than the Miller family. This is correct.
• Option 4: "The Miller family earns more than the Nelson family." This is the opposite of what we just concluded, so it's incorrect.

Answer:

(a): About 84% of the families in their county earn less than the Nelson family.

Part (b)