Don’t call me According to a study by Nielsen Mobile, “Teenagers ages $13$ to $17$ are by far the most prolific texters, sending $1742$ messages a month.” Mr. Williams, a high school statistics teacher, was skeptical about the claims in the article. So he collected data from his first-period statistics class on the number of text messages they had sent over the past $24$ hours. Here are the data:

a. Sunny was the student who sent $42$ text messages. What is Sunny’s percentile?

b. Find the number of text messages sent by Joelle, who is at the 16th percentile of the distribution.

Data values:

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

Now,

Sort the values of all the data in ascending order:

$$\begin{gathered} 0,0,0,1,1,3,3,5,5,7,8,8,9,14,25,25,26,29,42,44,52,72,92, \\ 98,118 \end{gathered}$$

Note that

$18$of the $25$data values are smaller than $42$

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

$$\begin{gathered} Percentile=\frac{n}{N}\times 100\% \\ =\times 100\%1825 \\ =0.72\times 100\% \\ =72\% \end{gathered}$$

Thus,

The percentile that corresponds with $42$text messages is the $72nd$percentile.

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,

Sort all the data values in ascending order:

$$\begin{gathered} 0,0,0,1,1,3,3,5,5,7,8,8,9,14,25,25,26,29,42,44,52,72,92, \\ 98,118 \end{gathered}$$

Note that

The data set consists of $25$ data values.

Such that

$N=25$

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

Then

$16th$ percentile has $16\%$ of the $25$ data values below it.

Calculate $16\%$ of $25$ data values:

$n=16\%\times N=\frac{16}{100}\times 25=\frac{4}{25}\times 25=4$

We have

$4$ out of $25$ data values are smaller than $16th$ percentile.

$16th$ percentile needs to be the $5th$ data value in the ascending order, and that is $1$

Thus,

The 16th percentile is $1$ text message.