Question

Run fast! As part of a student project, high school students were asked to

sprint $50$ yards and their times (in seconds) were recorded. A cumulative relative frequency graph of the sprint times is shown here.

a. One student ran the $50$ yards in $8$ seconds. Is a sprint time of $8$ seconds unusually slow?

b. Estimate and interpret the $20th$ percentile of the distribution.

Short answer

Part (a) Sprint time of $8$ seconds is not usually slow.

Part (b) Around $20\%$ of the students sprint the $50$ yards in less than $6.75$ seconds.

Step-by-step solution

Part (a) Step 1: Given information

The figure 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: Explanation

First

At $8$, draw a vertical line that intersects the horizontal axis.

Then

Draw a horizontal line through the point where the curve and the vertical line intersect.

Note that

The horizontal line roughly intersects the axis at $73$

That implies

Around $73$ percent of the children completed the $50-$yard dash in under eight seconds.

And

The remaining $27\%$ of pupils ran the $50-$yard dash in under 8 seconds.

Since

A percentage less than $5\%$ is considered to be small.

Thus,

Because $27$ percent cannot be deemed little, an $8-$second sprint time is not considered slow.

Part (b) Step 1: Explanation

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

That implies

The $20th$ percentile represents a data value with $20\%$ of the data values below it.

Now,

Draw the horizontal line intersecting the vertical axis at $20$

Then

Draw a vertical line through the curve and the horizontal line's intersection.

Note that the vertical line roughly intersects the horizontal axis at $6.75$

That implies

The $20th$ percentile is roughly at $6.75$ seconds.

Thus,

In less than $6.75$ seconds, around $20\%$ of all the students sprint $50$ yards.