Holt Algebra 2

11-Ext Normal Distributions11-Ext Normal Distributions

Holt Algebra 2

Lesson PresentationLesson Presentation

Holt Algebra 2

11-Ext Normal Distributions

Recognize normally distributed data.

Use the characteristics of the normal distribution to solve problems.

Objectives

Holt Algebra 2

11-Ext Normal Distributions

Standardized test results, like those used for college admissions, 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.

Holt Algebra 2

11-Ext Normal Distributions

The binomial distributions that you studied in Lesson 11-6 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.

Holt Algebra 2

11-Ext Normal Distributions

In a continuous probability distribution, the outcome can be any real number—for example, the time it takes to complete a 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.

Holt Algebra 2

11-Ext Normal Distributions

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.

Holt Algebra 2

11-Ext Normal Distributions

The figure shows the percent of data in a normal distribution that falls within a number of standard deviations from the mean.

Holt Algebra 2

11-Ext Normal Distributions

Addition shows the following:• About 68% lie within 1 standard deviation of the mean.• About 95% lie within 2 standard deviations of

the mean.• Close to 99.8% lie within 3 standard deviations

of the mean.

Holt Algebra 2

11-Ext Normal Distributions

Example 1A: Finding Normal Probabilities

The 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?

300 is 2 standard deviations from the mean, 500. Use the percents from the previous slide.

13.6% + 34.1% = 47.7%

Holt Algebra 2

11-Ext Normal Distributions

Example 1B: Finding Normal Probabilities

What is the probability that an SAT score is below 700?

Because the graph is symmetric, the left side of the graph shows 50% of the data.

50% + 47.7% = 97.7%

The probability that an SAT score is below 700 is 0.977, or 97.7%.

Holt Algebra 2

11-Ext Normal Distributions

Example 1C: Finding Normal Probabilities

What is the probability that an SAT score is less than 400 or greater than 600?

50% – 34.1% = 15.9 Percent of data > 600

Because the curve is symmetric, the probability that an SAT score is less than 400 or greater than 600 is about 2(15.9%), or 31.8%.

Holt Algebra 2

11-Ext Normal Distributions

Check It Out! Example 1

Use the information in Example 1 to answer the following.

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

50% – (34.1% + 13.6%) = 2.3% Percent of scores below 300.

The probability of a score above 300 is 1 – 2.3% or 97.7%.