Chapter 17, Problem 70E

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 again the college entrance exam discussed in Exercise 68.

a. Mary scored 83 points on the exam. Estimate the percentile in which this score places her.

b. Adam scored 45 points on the exam. Estimate the percentile in which this score places him.

c. If there are 250 students that scored 35 points or less on the exam, estimate the total number of students who took the exam.