Chapter 8 ExtensionNormal Distributions

Objectives

Recognize normally distributed data

Use the characteristics of the normal distribution to solve problems

Normal DistributionStandardized test results, like

those used for college admission, follow a normal distribution

Probability distributions can be based on either discrete or continuous data. Usually discrete data result from counting and continuous data result from measurement

Normal DistributionThe binomial

distribution were discrete probability distributions because there was a finite number of possible outcomes. The graph shows the probability distribution of the number of questions answered correctly when guessing on a true-false test.

Normal Distribution In a continuous probability distribution, the

outcome can be any real number – for example, the time it takes to complete at task

You may be familiar with the bell-shaped curve called the normal curve. A normal distribution is a function of the mean and standard deviation of a data set that assigns probabilities to intervals of real numbers associated with continuous random variables

Normal DistributionsThe probability assigned to a real-number

interval is the area under the normal curve in that interval. Because the area under the curve represents probability, the total area under the curve is 1

The maximum value of a normal curve occurs at the mean

The normal curve is symmetric about a vertical line through the mean

The normal curve has a horizontal asymptote at y = 0

Normal DistributionsThe figure shows the percent of data in a normal distribution

that falls within a number of standard deviations from the mean

Addition shows the following: About 68% of the data lies within 1 standard deviation of the mean About 95% of the data lie within 2 standard deviations of the mean Close to 99.8% of the data lie within 3 standard deviations of the mean

ExampleThe SAT is designed so that

scores are normally distributed with a mean of 500 and a standard deviation of 100. What percent of SAT scores are between 300 and 500?

Example (Continued)What is the probability that an

SAT score is below 700?

Example (Continued) What is the probability that an SAT score is less than

400 or greater than 600?

What is the probability that an SAT score is above 300?

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