Question

Here is a stem plot of the percent of residents aged $65$ and older in the $50$ states:

a. Find the percentile for Colorado, where $10.1\%$ of the residents are aged $65$ and older.

b. Rhode Island is at the $80th$ percentile of the distribution. Interpret this value. What percent of Rhode Island’s residents are aged $65$ and older?

Short answer

Part (a) Percentile for Colorado with $10.1\%$ of the residents is $8th$ percentile.

Part (b) $14\%$ of Rhode Island’s residents are $65$ or older and is at $80th$ percentile.

Step-by-step solution

Part (a) Step 1: Given information

Below is a graph of the percentage of residents aged $65$ and up.

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

Let

$n,$ the number of data values that are fewer than the data value of a single individual

$N,$ Total number of data values

Now,

In ascending order, write down all of the data values from the stem plot:

Where,

$7|0represents7.0$

$$\begin{gathered} 7.0,8.8,9.8,10.0,10.1,10.9,11.1,11.6,11.7,11.7,11.7,12.0,12.1, \\ 12.1,12.2,12.2,12.4,12.5,12.6,12.7,12.7,12.8,12.9,12.9,12.9,13.0, \\ 13.0,13.0,13.1,13.2,13.2,13.3,13.3,13.4,13.4,13.4,13.5,13.5,13.6, \\ 13.8,13.9,14.0,14.2,14.3,14.5,14.6,14.8,15.2,15.4,16.9. \end{gathered}$$

Note that

$4$ of the $50$ states have a percentage below $10.1\%$

The percentile is derived by dividing the entire number of data values by the number of data values fewer than the individual's data value.

$$\begin{gathered} Percentile=\frac{4}{50}\times 100\% \\ =0.08\times 100\% \\ =8\% \end{gathered}$$

Thus,

The percentile that corresponds to $10.1\%$ of the residents is $8$ percentile.

Part (b) Step 1: Calculation

Let

$n,$ The number of data values that are less than the data value of an individual.

$N,$ Total number of data values

Now,

In ascending order, write down all of the data values from the stem plot:

Where,

$7|0represents7.0$

$$\begin{gathered} 7.0,8.8,9.8,10.0,10.1,10.9,11.1,11.6,11.7,11.7,11.7,12.0,12.1, \\ 12.1,12.2,12.2,12.4,12.5,12.6,12.7,12.7,12.8,12.9,12.9,12.9,13.0, \\ 13.0,13.0,13.1,13.2,13.2,13.3,13.3,13.4,13.4,13.4,13.5,13.5,13.6, \\ 13.8,13.9,14.0,14.2,14.3,14.5,14.6,14.8,15.2,15.4,16.9 \end{gathered}$$

Note that

In the sample, there are total $50$ states (or $50$ data values).

Such that

$N=50$

The $xth$ percentile represents a data value with $x$ percent of the data values below it.

When Rhode Island is in the $80th$ percentile, it means that $80\%$ of the data values are below it.

Calculate $80\%$ of $50$ data values:

$n=80\%\times N=\frac{80}{100}\times 50=\frac{4}{5}\times 50=40$

We have

The $80th$ percentile is reached by $40$ out of $50$ data values. The data value for the $80th$ percentile must be the $41st$ in ascending order, which is $14.0$

Thus,

In Rhode Island, $14\%$ of residents are $65$ or older and are at the $80th$ percentile.