11-Ext
11-ExtNormal
NormalDistributions
Distributions
Lesson Presentation
Holt
Algebra
Holt
Algebra
22
11-Ext
Normal Distributions
Objectives
Recognize normally distributed data.
Use the characteristics of the normal
distribution to solve problems.
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 Distributions
The 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
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
Percent of data > 600
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%.
Holt Algebra 2