Chapter 17, Problem 72E

Percentiles. The pth percentile of a sorted data set is a number $x_p$ such that p% of the data fall at or below $x_p$ and $\left(100-p\right)$ of the data fall at or above $x_p$. (For details, see Chapter 15, Section 15.2.) For normally distributed data sets, there are detailed statistical tables that give the location of the pth percentile for every possible p between 1 and 99. The following table is an abbreviated version giving the approximate location of some of the more frequently used percentiles in a normal distribution with mean $\mu$ and standard deviation $\sigma$. For approximately normal distributions, Table 17-19 can be used to estimate these percentiles.

Table 17-9

Percentile	Approximate location	Percentile	Approximate location
99th	$\mu +2.33\sigma$	1st	$\mu -2.33\sigma$
95th	$\mu +1.65\sigma$	5th	$\mu -1.65\sigma$
90th	$\mu +1.28\sigma$	10th	$\mu -1.28\sigma$
80th	$\mu +0.84\sigma$	20th	$\mu -0.84\sigma$
75th	$\mu +0.675\sigma$	25th	$\mu -0.675\sigma$
70th	$\mu +0.52\sigma$	30th	$\mu -0.52\sigma$
60th	$\mu +0.25\sigma$	40th	$\mu -0.25\sigma$
50th	$\mu$

In Exercises 67 through 73, you should use the table to make your estimates.

Consider a normal distribution with mean $\mu =0$ and standard deviation $\sigma =1$.

a. Find the 90th percentile (rounded to two decimal places).

b. Find the 10th percentile (rounded to two decimal places).

c. Find the 80th percentile (rounded to two decimal places).

d. Find the 20th percentile (rounded to two decimal places).

e. Suppose that you are given that the 85th percentile is approximately 1.04. Find the 15th percentile.