11.3 Use Normal Distributions

Normal distribution

A probability distribution modeled by a bell shaped curve that is symmetric about the mean

Normal curve

The bell shaped curve that models a normal distribution

Standard normal distribution

The normal distribution with mean 0 and standard deviation 1

z-score

The z-value for a particular x value and is the number of standard deviations the x value lies above or below the mean

AREAS UNDER A NORMAL CURVE

A normal distribution with mean and standard deviation s has these properties:

· The total area under the related normal curve is _1_.

· About _68_ % of the area lies within 1 standard deviation of the mean.

· About _95_ % of the area lies within 2 standard deviations of the mean.

· About _99.7_ % of the area lies within 3 standard deviations of the mean.

Example 1

Find a normal probability

A normal distribution has mean x and standard deviation s. For a randomly selected x-value from the distribution, find P( £ x £ + 3s)

Solution

The probability that a randomly selected x-value lies between ____ and ___+ 3s_ is the shaded area under the normal curve. Therefore:

P( £ x £ + 3s) = _0.34_ + _0.135_ + _0.0235_

= _0.4985_

Example 2

Interpret normally distributed data

Math Scores The math scores of the 2004 SAT exam are normally distributed with a mean of 518 and a standard deviation of 114.

a. About what percent of the test-takers have scores between 518 and 746?

b. About what percent of the test-takers have scores less than 404?

Solution

a. The scores of 518 and 736 represent _two_ standard deviations to the _right_ of the mean. So, the percent of test-takers with scores between 518 and 736 is _34_ % + _13.5_ % = _47.5_ %.

b. A score of 404 is _one_ standard deviation to the _left_ of the mean. So, the percent of scores less than 404 is

_13.5_ % + _2.35_ % + _0.15_ % = _16_ %.

Example 3

Use a z-score and the standard normal table

Height A survey of a group of women found that the height of the women is normally distributed with a mean height of 64.5 inches and a standard deviation of 2.5 inches. Find the probability that a woman is at most 58 inches tall.

Solution

1. Find the z-score corresponding to an x-value of 58.

= = _-2.6_

2. Use the standard normal table to find P(x £ 58) = P(z £ _-2.6_). The table shows that P(z £ _-2.6_) = _0.0047_. So, the probability that a woman is at most 58 inches tall is about _0.0047_.

z	.0	.1	.2	.3	.4
-2	.0228	.0179	.0139	.0107	.0082
2	.9772	.9821	.9861	.9893	.9918
z	.5	.6	.7	.8	.9
-2	.0062	.0047	.0035	.0026	.0019
2	.9938	.9953	.9965	.9974	.9981