Chapter 2.2, Problem 64E

To determine

85^th percentile of Djokovic’s first − serve speeds.

Answer to Problem 64E

85^th percentile is 117.2 mph.

Explanation of Solution

Given information:

Mean speed, $\mu =112\text{mph}$

Standard deviation, $\sigma =5\text{mph}$

Calculations:

The data value represented by the x^th percentile has x% of the data values below it.

That implies

The data value represented by the 85^th percentile has 85% of the data values below it.

Now,

Find the z − score corresponding the probability of 85% (or 0.85) in the normal probability table of the appendix.

Note that

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

Then

The corresponding z − score,

  $z=1.04$

In order to find x^th 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,

85^thpercentile will be 117.2 mph.