Chapter 17, Problem 67E

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.

The distribution of weights for six-month baby boys has mean $\mu =8.16$ kg and standard deviation $\sigma =0.95$ kg. (see Exercise 53).

a. Suppose that a six-month-old boy weighs in the 95^th percentile of his age group. Estimate his weight in kilograms rounded to two decimal places.

b. Suppose that a six-month-old baby boy weighs in the 20^th percentile of his age group. Estimate his weight in kilograms rounded to two decimal places.