Question

Making money $(2.1)$ The parallel dot plots show the total family income of randomly chosen individuals from Indiana ($38$ individuals) and New Jersey ($44$ individuals). Means and standard deviations are given below the dot plots.

Consider individuals in each state with total family incomes of $\$95,000$ Which individual has a higher income, relative to others in his or her state? Use percentiles and $z-$scores to support your answer.

Short answer

In comparison to others in his or her state, an Indiana resident enjoys a better income.

Step-by-step solution

Given information

Concept

The formula used:$z=\frac{x-\mu }{\sigma }$

Calculation

For Indiana:

Calculate the $z-$score:

$z=\frac{x-\mu }{\sigma }=\frac{95,000-47,400}{29,400}=\frac{47,600}{29,400}\approx 1.62$

For New Jersey:

Calculate the $z-$score:

$z=\frac{x-\mu }{\sigma }=\frac{95,000-58,100}{41,900}=\frac{36,900}{41,900}\approx 0.88$

Compare the $z-$scores:

When z scores are compared, it shows that a person with a higher $z-$score has a greater income than others in his or her state.

As a result, an Indiana resident has a higher income than others in his or her state.