Shoes How many pairs of shoes does a typical teenage boy own? To find out,

two AP® Statistics students surveyed a random sample of $20$ male students from their large high school. Then they recorded the number of pairs of shoes that each boy owned. Here is a dot-plot of the data:

a. Find the percentile for Jackson, who reported owning $22$ pairs of shoes.

b. Raul’s reported number of pairs of shoes is at the ${45}^{th}$ percentile of the distribution. Interpret this value. How many pairs of shoes does Raul own?

The dot plot below illustrates a random sample of $20$ male students who own a certain number of pairs of shoes.

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

Let

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

$N,$Total number of data values

Note that

$17$of the $20$dots in the dot plot are to the left of $22$

Thus,

$17$of the $20$people in the survey have fewer than $22$pairs of shoes.

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.

$$\begin{gathered} Percentile=\frac{n}{N}\times 100\% \\ =\frac{17}{20}\times 100\% \\ =0.85\times 100\% \\ =85\% \end{gathered}$$

Thus,

The percentile that corresponds with $22$pairs of shoes is the ${85}^{th}$percentile.

Let

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

$N,$Total number of data values

Now,

Write down all of the data values from the dot plot in ascending order, as seen on the horizontal axis in the form of dots.

$4,5,5,5,6,7,7,7,7,8,10,10,10,10,11,12,14,22,35,38$

Note that

There are a total of $20$data values in the sample.

Such that

$N=20$

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

The $45th$percentile has $45$percent of the data values below it, so Raul's number of shoe pairs is at the$45th$ percentile.

Calculate $45\%$of $20$data values:

$n=45\%\times N=\frac{45}{100}\times 20=\frac{9}{20}\times 20=9$

We have

$9$ out of $20$ data values below $45$ the percentile.

The $45th$ percentile should be the tenth data value in ascending order, which is $8$

Thus,

Raul owns $8$ pairs of shoes and is in the ${45}^{th}$ percentile.