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Chapter 6
Probability Distributions
Section 6.2
Probabilities for Bell-Shaped Distributions
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Normal Distribution
The normal distribution is symmetric, bell-shaped and
characterized by its mean  and standard deviation  .
 The normal distribution is the most important
distribution in statistics.
 Many distributions have an approximately normal
distribution.
 The normal distribution also can approximate
many discrete distributions well when there are a
large number of possible outcomes.
 Many statistical methods use it even when the
data are not bell shaped.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Normal Distribution
Normal distributions are
 Bell shaped
 Symmetric around the mean
The mean ( ) and the standard deviation ( ) completely
describe the density curve.
 Increasing/decreasing  moves the curve along the
horizontal axis.
 Increasing/decreasing  controls the spread of the
curve.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Normal Distribution
Within what interval do almost all of the men’s heights
fall? Women’s height?
Figure 6.4 Normal Distributions for Women’s Height and Men’s Height. For each different
combination of  and  values, there is a normal distribution with mean  and standard
deviation  . Question: Given that  = 70 and  = 4, within what interval do almost all of the
men’s heights fall?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Normal Distribution: 68-95-99.7 Rule for
Any Normal Curve
≈ 68% of the observations fall within one standard deviation of the mean.
≈ 95% of the observations fall within two standard deviations of the mean.
≈ 99.7% of the observations fall within three standard deviations of the mean.
Figure 6.5 The Normal Distribution. The probability equals approximately 0.68 within
1 standard deviation of the mean, approximately 0.95 within 2 standard deviations,
and approximately 0.997 within 3 standard deviations. Question: How do these
probabilities relate to the empirical rule?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example : 68-95-99.7% Rule
Heights of adult women can be approximated by a normal
distribution,
  65
inches;
  3.5
inches
68-95-99.7 Rule for women’s heights:
68% are between 61.5 and 68.5 inches
 [     65  3.5]
95% are between 58 and 72 inches
 [   2  65  2(3.5)  65  7]
99.7% are between 54.5 and 75.5 inches
 [   3  65  3(3.5)  65  10.5]
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Z-Scores and the Standard Normal
Distribution
The z-score for a value x of a random variable is the
number of standard deviations that x falls from the mean.
z
x 

A negative (positive) z-score indicates that the value is
below (above) the mean.
Z-scores
can be used to calculate the probabilities of a

normal random variable using the normal tables in Table A
in the back of the book.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Z-Scores and the Standard Normal
Distribution
A standard normal distribution has mean
 0
standard deviation   1 .
and
When a random variable has a normal distribution and
its values are converted to z-scores by subtracting the
mean and dividing by the standard deviation, the z-scores
follow the standard normal distribution.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Table A: Standard Normal Probabilities
Table A enables us to find normal probabilities.
 It tabulates the normal cumulative probabilities
falling below the point   z .
To use the table:
 Find the corresponding z-score.
 Look up the closest standardized score (z) in the
table.
 First column gives z to the first decimal place.
 First row gives the second decimal place of z.
 The corresponding probability found in the body of
the table gives the probability of falling below the
z-score.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Using Table A
Find the probability that a normal random variable takes
a value less than 1.43 standard deviations above ;
P( z  1.43)  0.9236
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Using Table A
Figure 6.7 The Normal Cumulative Probability, Less than z Standard Deviations
above the Mean. Table A lists a cumulative probability of 0.9236 for z  1.43, so
0.9236 is the probability less than 1.43 standard deviations above the mean of any
normal distribution (that is, below   1.43 ). The complement probability of 0.0764 is
the probability above   1.43 in the right tail.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Using Table A
Find the probability that a normal random variable
assumes a value within 1.43 standard deviations of  .
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
Probability below 1.43  0.9236

Probability below 1.43  0.0764

P(1.43  z  1.43)  0.9236  0.0764  0.8472
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
How Can We Find the Value of z for a
Certain Cumulative Probability?
To solve some of our problems, we will need to find the
value of z that corresponds to a certain normal cumulative
probability.
To do so, we use Table A in reverse.
 Rather than finding z using the first column (value
of z up to one decimal) and the first row (second
decimal of z).
 Find the probability in the body of the table.
 The z-score is given by the corresponding values
in the first column and row.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
How Can We Find the Value of z for a
Certain Cumulative Probability?
Example: Find the value of z for a cumulative probability of 0.025.
Look up the cumulative probability of 0.025 in the body of Table A.
A cumulative probability of 0.025
corresponds zto 1.96 .
Thus, the probability that a normal
random variable falls at least
1.96 standard deviations
below the mean is 0.025.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
SUMMARY: Using Z-Scores to Find Normal
Probabilities or Random Variable x Values
 If we’re given a value x and need to find a probability,
convert x to a z-score using z  ( x   ) /  , use a table of
normal probabilities (or software, or a calculator) to get a
cumulative probability and then convert it to the probability
of interest
 If we’re given a probability and need to find the value of
x , convert the probability to the related cumulative
probability, find the z-score using a normal table (or
software, or a calculator), and then evaluate x    z .
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: Comparing Test Scores That
Use Different Scales
Z-scores can be used to compare observations from
different normal distributions.
Picture the Scenario:
There are two primary standardized tests used by college
admissions, the SAT and the ACT.
You score 650 on the SAT which has   500 and   100
and 30 on the ACT which has   21.0 and   4.7.
How can we compare these scores to tell which score is
relatively higher?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Using Z-scores to Compare Distributions

Compare z-scores:
SAT: z 
650  500
1.5
100
30  21
 1.91
ACT: z 

4.7
Since your z-score is greater for the ACT, you

performed
relatively better on this exam.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.