© 2008 Pearson Addison-Wesley. All rights reserved 13-5-1 Chapter 1 Section 13-5 The Normal...

© 2008 Pearson Addison-Wesley. All rights reserved

13-5-1

Chapter 1

Section 13-5The Normal Distribution


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The Normal Distribution

• Discrete and Continuous Random Variables

• Definition and Properties of a Normal Curve

• A Table of Standard Normal Curve Areas

• Interpreting Normal Curve Areas


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Discrete and Continuous Random Variables

A random variable that can take on only certain fixed values is called a ___________________. A variable whose values are not restricted in this way is a _______________.


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Definition and Properties of a Normal Curve

A _____________ is a symmetric, bell-shaped curve. Any random variable whose graph has this characteristic shape is said to have a ________________.

On a normal curve, if the quantity shown on the horizontal axis is the number of standard deviations from the mean, rather than values of the random variable itself, then we call the curve the _____________________________.


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Normal Curves

0

S

B

A

C


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Properties of Normal Curves

The graph of a normal curve is bell-shaped and symmetric about a vertical line through its center.

The mean, median, and mode of a normal curve are all equal and occur at the center of the distribution.

Empirical Rule About 68% of all data values of a normal curve lie within 1 standard deviation of the mean (in both directions), and about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations.


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Empirical Rule


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Example: Applying the Empirical Rule

Suppose that 280 sociology students take an exam and that the distribution of their scores can be treated as normal. Find the number of scores falling within 2 standard deviations of the mean.

Solution


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A Table of Standard Normal Curve Areas

To answer questions that involve regions other than 1, 2, or 3 standard deviations of the mean we can refer to the table on page 829 or other tools such as a computer or calculator.

The table gives the fraction of all scores in a normal distribution that lie between the mean and z standard deviations from the mean. Because of symmetry, the table can be used for values above the mean or below the mean.


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Example: Applying the Normal Curve Table

Use the table to find the percent of all scores that lie between the mean and the following values.

a) 1.5 standard deviation above the mean

b) 2.62 standard deviations below the mean


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Solution


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Find the total area indicated in the region in color below.

Solution xz = –1.7 z = 2.55


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Solution z = 2.63z = .61


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Solution z = 2.14


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Interpreting Normal Curve Areas

In a standard normal curve, the following three quantities are equivalent.


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Example: Probability with Volume

The volumes of soda in bottles from a small company are distributed normally with mean 12 ounces and standard deviation .15 ounce. If 1 bottle is randomly selected, what is the probability that it will have more than 12.33 ounces?

Solution


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Example: Finding z-Values for Given Areas

Assuming a normal distribution, find the z-value meeting the condition that 39% of the area is to the right of z.

Solution


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Example: Finding z-Values for Given Areas

Assuming a normal distribution, find the z-value meeting the condition that 76% of the area is to the left of z.

Solution

© 2008 Pearson Addison-Wesley. All rights reserved 13-5-1 Chapter 1 Section 13-5 The Normal...

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