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Chapter
37
The Normal
Probability
Distribution
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Pearson Prentice Hall. All rights reserved
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Section 7.2 The Standard Normal Distribution
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The table gives the area under the standard normal curve for
values to the left of a specified Z-score, zo, as shown in the
figure.
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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 left of z = -0.38 is 0.3520.
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Area under the normal curve to the
right of zo = 1 – Area to the left of zo
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EXAMPLE
Finding the Area Under the Standard Normal Curve
Find the area under the standard normal curve to the right of Z = 1.25.
Area right of 1.25 = 1 – area left of 1.25
= 1 – 0.8944
= 0.1056 © 2010 Pearson Prentice Hall. All rights reserved
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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.
Area between -1.02 and 2.94 = (Area left of z = 2.94) – (area left of z = -1.02)
= 0.9984 – 0.1539
= 0.8445
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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 0.7157.
The z-score such that the area to the left of the z-score is 0.7157 is z = 0.57.
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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.
The area left of the z-score is 1 – 0.3021 = 0.6979.
The approximate z-score that corresponds to an area of 0.6979 to the left (0.3021
to the right) is 0.52. Therefore, z = 0.52.
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EXAMPLE
Finding a z-score from a Specified Area
Find the z-scores that separate the middle 80% of the area under the normal curve
from the 20% in the tails.
Area = 0.8
Area = 0.1
Area = 0.1
z1 is the z-score such that the area left is 0.1, so z1 = -1.28.
z2 is the z-score such that the area left is 0.9, so z2 = 1.28.
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The notation zα (prounounced “z sub alpha”) is the z-score such that
the area under the standard normal curve to the right of zα is α.
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EXAMPLE
Finding the Value of z
Find the value of z0.25
We are looking for the z-value such that the area to the right of the z-value is
0.25. This means that the area left of the z-value is 0.75.
z0.25 = 0.67
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Notation for the Probability of a Standard Normal Random
Variable
P(a < Z < b)
represents the probability a standard
normal random variable is between
a and b
P(Z > a)
represents the probability a standard
normal random variable is greater
than a.
P(Z < a)
represents the probability a standard
normal random variable is less than a.
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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)
(a) P(Z < -0.23) = 0.4090
(b) P(Z > 1.93) = 0.0268
(c) P(0.65 < Z < 2.10) = 0.2399
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For any continuous random variable, the probability of observing a specific
value of the random variable is 0. For example, for a standard normal
random variable, P(a) = 0 for any value of a. This is because there is no area
under the standard normal curve associated with a single value, so the
probability must be 0. Therefore, the following probabilities are equivalent:
P(a < Z < b) = P(a < Z < b) = P(a < Z < b) = P(a < Z < b)
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