Question

length of songs in seconds
142, 156, 171, 174, 210, 222, 222, 227, 242
owen adds one more song to the playlist and the percentile rank of 210 becomes 60.
what could be the length of the new song?
select all possible options.

Explanation:

Step1: Recall percentile formula

The formula for the percentile rank \(P\) of a value \(x\) in a data - set of size \(n\) (after arranging the data in ascending order) is \(P=\frac{k}{n}\times100\), where \(k\) is the number of values less than or equal to \(x\). Initially, there are \(n = 9\) data - points. After adding one more song, \(n=10\). We know that the percentile rank of \(210\) is \(60\), so using the formula \(60=\frac{k}{10}\times100\), we can solve for \(k\).

Step2: Solve for \(k\)

\[

$$\begin{align*} 60&=\frac{k}{10}\times100\\ \frac{60}{100}&=\frac{k}{10}\\ k& = 6 \end{align*}$$

\]
This means that when the new song is added, there should be \(6\) values less than or equal to \(210\) in the new data - set of \(n = 10\) values. Initially, there are \(4\) values less than \(210\) (\(142,156,171,174\)). So the new song's length must be such that when it is added to the data - set, the number of values less than or equal to \(210\) becomes \(6\). So the new song's length must be less than \(210\).

Answer:

A. 178, C. 164, E. 140