Applications

Example 1

Assume that heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the height that correspsonds to the 75th percentile.

Solution .

First sketch a graph showing approximately where the 75th percentile is. (75% of the area will be to the left of the 75th percentile and 25% will be to the right.) Note that your sketch does not need to be as accurate as the one below. Your line for the 75th percentile should certainly be to the right of the mean.

On the z-scale our picture would look like the following (Again your sketch does not have to be as accurate as the computer drawn image below.)

Note that the line indicating the 75th percentile is drawn to the right of the mean which is zero for the standard normal.

We need to find the value of z at the 75th percentile line drawn in navy blue on the graph. The area of .75 is made up of two components, the area to the left of the mean of zero and the area between the mean of zero and the z value corresponding to the 75th percentile. Since .75 = .5000 + .2500, we must find the value of z for which the area from zero to that value is .2500. Using the z table, we find that z = .67

We must now convert this z score to the corresponding x score using

Letting z = .67, , and , we get

After multiplying both side by 2.5, we have 1.675 = x - 63.6.

Adding 63.6 to both sides, we get x = 65.275 or 65.3 inches.

The 75th percentile is 65.3 inches.

SUMMARY

To find the nth percentile for a nonstandard normal distribution

- Find the z score for the nth percentile.

- Convert this z score to an x score using

and a little algebra.

To convert the z score to an x score you may also use the formula which is used in the text.

Example 2

Assume that the heights of men are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches. Find the 15th percentile.

Solution .

We first first find the z score corresponding to the 15th percentile.

Since the area to the left of the 15th percentile is .15, the area from z = 0 to the z score corresponding to the 15th percentile must be .5000 - .1500 = .3500. Using the table, the z value nearest an area of .3500 is 1.04. Since our z must be negative, z = -1.04.

To convert this to an x score use with z = -1.04, , and We get

Multiplying by 2.8, we get -2.912 = x - 69.

Solving this we have x = 66.008 or x = 66.1 inches

The 15th percentile for the heights of men is 66.1 inches.