Question

Household incomes The cumulative relative frequency graph describes the distribution of median household incomes in the $50$ states in a recent year.

a. The median household income in North Dakota that year was $\$55,766$Is North Dakota an unusually wealthy state?

b. Estimate and interpret the $90th$ percentile of the distribution.

Short answer

Part (a) North Dakota is not an unusually wealthy state.

Part (b) The median household income of $90\%$ of all the states is less than $\$64,000$

Step-by-step solution

Part (a) Step 1: Given information

The figure is

Part (a) Step 2: Concept

The percentile of an individual is the percentage of the distribution that is less than the data value of the individual.

Part (a) Step 3: Explanation

First

Because $\$55,766$ is about fifty-six thousand dollars, draw a vertical line intersecting the horizontal axis at $56$

Then

Draw a horizontal line through the curve and the vertical line's intersection.

Note that

At $68.4$, the horizontal line intersects the vertical axis roughly.

That implies

Around $68.4\%$ of the states have lower median household incomes.

And

The rest $31.6\%$ of the states have higher median household income.

Since

A percentage less than $5\%$ is considered to be small.

As a result, North Dakota is not typically wealthy, as its population of $31.6$ percent cannot be regarded as insignificant.

Part (b) Step 1: Explanation

The data value represented by the $Xth$ percentile has $x\%$ of the data values below it.

That implies

The $90th$ percentile represents a data value with $90\%$ of the data values below it.

Now,

Draw a horizontal line at $90$ degrees to the vertical axis.

Then

Draw a vertical line through the curve and the horizontal line's intersection.

Note that

At $64$, the vertical line touches the horizontal axis roughly.

That implies

The $90th$ percentile is roughly at $\$64,000$

Thus,

The median household income of 90% of all the states is less than $\$64,000$