Transcript 13-5-1

Chapter 1
Section 13-5
The Normal Distribution
13-5-1
© 2008 Pearson Addison-Wesley. All rights reserved
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
13-5-2
© 2008 Pearson Addison-Wesley. All rights reserved
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
_______________.
13-5-3
© 2008 Pearson Addison-Wesley. All rights reserved
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
_____________________________.
13-5-4
© 2008 Pearson Addison-Wesley. All rights reserved
Normal Curves
B
A
S
C
0
13-5-5
© 2008 Pearson Addison-Wesley. All rights reserved
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.
13-5-6
© 2008 Pearson Addison-Wesley. All rights reserved
Empirical Rule
13-5-7
© 2008 Pearson Addison-Wesley. All rights reserved
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
13-5-8
© 2008 Pearson Addison-Wesley. All rights reserved
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.
13-5-9
© 2008 Pearson Addison-Wesley. All rights reserved
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
13-5-10
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Applying the Normal Curve
Table
Solution
13-5-11
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Applying the Normal Curve
Table
Find the total area indicated in the region in
color below.
Solution
z = –1.7
x
z = 2.55
13-5-12
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Applying the Normal Curve
Table
Find the total area indicated in the region in
color below.
Solution
z = .61
z = 2.63
13-5-13
© 2008 Pearson Addison-Wesley. All rights reserved
Example: Applying the Normal Curve
Table
Find the total area indicated in the region in
color below.
z = 2.14
Solution
13-5-14
© 2008 Pearson Addison-Wesley. All rights reserved
Interpreting Normal Curve Areas
In a standard normal curve, the following three
quantities are equivalent.
13-5-15
© 2008 Pearson Addison-Wesley. All rights reserved
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
13-5-16
© 2008 Pearson Addison-Wesley. All rights reserved
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
13-5-17
© 2008 Pearson Addison-Wesley. All rights reserved
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
13-5-18
© 2008 Pearson Addison-Wesley. All rights reserved