Question

Foreign-born residents The cumulative relative frequency graph shows the distribution of the percent of foreign-born residents in the $50$ states.

a. Estimate the interquartile range (IQR) of this distribution. Show your method.

b. What is the percentile for Arizona, which had $15.1\%$ foreign-born residents that year?

c. Explain why the graph is fairly flat between $20\%$ and $27.5\%$

d. Draw the histogram that corresponds to this graph.

Short answer

Part (a) The value is $9.00$

Part (b) The percentile is $84$

Part (c) The cumulative relative frequency for these values is growing between $20\%$ and $27.5$ percent.

Part (d) The histogram is

Step-by-step solution

Part (a) Step 1: Given information

The plot 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: Calculation

The third quartile and first quartile, according to the plot, are $12.75$ and $3.75$ respectively.

Thus, the interquartile range is:

$$\begin{gathered} IQR=Q_3-Q_1 \\ =12.75-3.75 \\ =9.00 \end{gathered}$$

Thus, the required value is $9.00$

Part (b) Step 1: Explanation

Create a vertical line that cuts the $x-$axis at $15.1$ to calculate the required percentage. Create a horizontal line to connect the vertical and horizontal lines. Both lines overlap at the $84th$ percentile, as can be seen.

Thus, the required percentile is $84$

Part (c) Step 1: Explanation

Because of the minor rise in frequency for these values, the cumulative relative frequency for these values is growing between $20\%$ and $27.5$ percent.

Part (d) Step 1: Explanation

The histogram could be constructed as: