Transcript Chapter 7 The Normal Probability Distribution 7.2
Chapter 7 The Normal Probability Distribution
7.2 The Standard Normal Distribution
Properties of the Normal Density Curve
7. The Empirical Rule: About 68% of the area under the graph is between -1 and 1; about 95% of the area under the graph is between -2 and 2; about 99.7% of the area under the graph is between -3 and 3.
The table gives the area under the standard normal curve for values to the left of a specified Z-score,
z
o , as shown in the figure.
EXAMPLE
Finding the Area Under the Standard Normal Curve
Find the area under the standard normal curve to the left of
Z
= -0.38.
Area under the normal curve to the right of
z
o
= 1 – Area to the left of
z
o
EXAMPLE
Finding the Area Under the Standard Normal Curve
Find the area under the standard normal curve to the right of
Z
= 1.25.
EXAMPLE
Finding the Area Under the Standard Normal Curve
Find the area under the standard normal curve between
Z
= -1.02 and
Z
= 2.94.
EXAMPLE
Finding a Z-score from a Specified Area to the Left
Find the
Z
-score such that the area to the left of the
Z
-score is 0.68.
EXAMPLE
Finding a Z-score from a Specified Area to the Right
Find the
Z
-score such that the area to the right of the
Z
-score is 0.3021.
EXAMPLE
Finding a Z-score
Find the
Z
-scores that separate the middle 92% of the area under the normal curve from the 8% in the tails.
EXAMPLE
Finding the Value of z
Find the value of
z
0.25
Notation for the Probability of a Standard Normal Random Variable
P
(
a < Z
<
b
) standard between represents the probability a normal random variable is
a
and
b P
(
Z > a
) standard greater represents the probability a normal random variable is than
a
.
P
(
Z
<
a
) standard less than represents the probability a normal random variable is
a
.
EXAMPLE
Finding Probabilities of Standard Normal Random Variables
Find each of the following probabilities: (a)
P
(
Z
< -0.23) (b)
P
(
Z
> 1.93) (c)
P
(0.65 <
Z
< 2.10)