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• How do I use
normal
distributions in
finding
probabilities?
Use Normal Distributions
Standard Deviation of a Data Set
A normal distribution with mean x and standard deviation
s has these properties:
• The total area under the related normal curve is ____.
1
• About ___%
68 of the area lies within 1 standard
deviation of the mean.
• About ___%
95 of the area lies within 2 standard
deviation of the mean.
• About _____%
99.7 of the area lies within 3 standard
deviation of the mean.
34%
68%
95%
99.7%
34%
13.5%
2.35%
13.5%
2.35%
0.15%
– 2s
–s
+ 3s
– 3s
+ 2s
+ 3s
+s
+ 2s
–s
+s
– 2s
x
x
x
x
x
x
x
x
x
x
x
x
x
x
– 3s
0.15%
Use Normal Distributions
+s
+ 2s
+ 3s
x
–s
x
– 2s
x
x
x
x
Px  s  x  x  2s 
x
A normal distribution has a mean
x and standard deviation s. For a
randomly selected x-value from the
distribution, find
– 3s
Example 1 Find a normal probability
Solution
The probability that a randomly selected x-value lies
between _______
x  2s is the shaded area
x  s and _________
under the normal curve. Therefore:
Px  s  x  x  2s   _____
0.34  _____
0.34  ______
0.135
 _______
0.815
Use Normal Distributions
Checkpoint. Complete the following exercise.
1. A normal distribution has mean x and
standard deviation s. For a randomly
selected x-value from the distribution,
find P x  x  s .
+s
+ 2s
+ 3s
x
–s
x
x
x
x
x
x
0.50  0.34  0.16
– 2s

– 3s

Use Normal Distributions
Example 2 Interpret normally distributed data
The math scores of an exam for the
state of Georgia are normally
distributed with a mean of 496 and a
standard deviation of 109. About what
percent of the test-takers received
169
scores between 387 and 605?
278 387 496 605 714 823
Solution
The scores of 387 and 605 represent ____
one standard
deviation on either side of the mean. So the percent of
test-takers with scores between 387 and 605 is
___
34 %  ____
34 %  ____
68 %
Use Normal Distributions
Checkpoint. Complete the following exercise.
2. In Example 2, what percent of
the test-takers received scores
between 496 and 714?
34%
13.5%
169 278 387 496 605 714 823
34  13.5 47.5%
Use Normal Distributions
Example 3 Use a z-score and the standard normal table
In Example 2, find the probability that a randomly
selected test-taker received a math score of at most 630?
Solution
Sep 1 Find the z-score corresponding to an x-value of 630.
630  496
z

 _____
1.2
s
109
xx
__________
__________
_
Use Normal Distributions
Example 3 Use a z-score and the standard normal table
In Example 2, find the probability that a randomly
selected test-taker received a math score of at most 630?
Solution
Sep 2 Use the standard normal table to find
Px  630  Pz  ___
1.2 
z
.0
.1
.2
3
.0013
.0010
.0007
2
.0228
.0179
.0139
1
.1587
.1357
.1151
0
.5000
.4602
.4207
0
.5000
.5398
.5793
1
.8413
.8643
.8849
The table shows that
P(z < ____)
1.2 = ._______.
8849
So, the probability
that a randomly
selected test-taker
received a math score
of at most 630 is
about ________.
.8849
Use Normal Distributions
Checkpoint. Complete the following exercise.
3. In Example 3, find the probability that a
randomly selected test-taker received a math
score of at most 620?
620  496
z

s
109
xx
Px  620  Pz  1.1
 1.1
 0.8643