Question

Hit an ace! Refer to Exercise $62$ Find the 85th percentile of Djokovic’s first-serve speeds.

Short answer

$85th$ percentile is $117.2$ mph.

Step-by-step solution

Given information

Mean speed,$\mu =112mph$

Standard deviation,$\sigma =5mph$

Concept

The formula used:$z=\frac{x-\mu }{\sigma }$

Calculation

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

That implies

The $85th$percentile represents a data value with$85$ percent of the data values below it.

Now,

In the normal probability table of the appendix, find the $z-$score that corresponds to a probability of $85$ percent (or $0.85$).

Note that

The closest probability would be $0.8508$ which lies in row $1.0$ and in column $.04$ of the normal probability table.

Then

The corresponding $z-$score,

$z=1.04$

In order to find the $xth$ percentile, calculate the $z-$score

$z=\frac{x-\mu }{\sigma }=\frac{x-112}{5}$

Equate the above two expressions:

$\frac{x-112}{5}=1.04$

Multiply both sides by $5:$

$x-112=5.2$

Add $112$to both sides:

$x=5.2+112$

That becomes

$x=117.2$

Thus,

$85th$percentile will be $117.2$mph.