Chapter
37
The Normal
Probability
Distribution
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Section 7.1 Properties of the Normal Distribution
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EXAMPLE
Illustrating the Uniform Distribution
Suppose that United Parcel Service is supposed to deliver a
package to your front door and the arrival time is somewhere
between 10 am and 11 am. Let the random variable X
represent the time from 10 am when the delivery is supposed
to take place. The delivery could be at 10 am (x = 0) or at 11
am (x = 60) with all 1-minute interval of times between x = 0
and x = 60 equally likely. That is to say your package is just as
likely to arrive between 10:15 and 10:16 as it is to arrive
between 10:40 and 10:41. The random variable X can be any
value in the interval from 0 to 60, that is, 0 < X < 60. Because
any two intervals of equal length between 0 and 60, inclusive,
are equally likely, the random variable X is said to follow a
uniform probability distribution.
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The graph below illustrates the properties for the “time”
example. Notice the area of the rectangle is one and the graph
is greater than or equal to zero for all x between 0 and 60,
inclusive.
Because the area of a
rectangle is height
times width, and the
width of the rectangle
is 60, the height must
be 1/60.
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Values of the random variable X less than 0 or greater than
60 are impossible, thus the equation must be zero for X
less than 0 or greater than 60.
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The area under the graph of the density function over an
interval represents the probability of observing a value of the
random variable in that interval.
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EXAMPLE
Area as a Probability
The probability of choosing a time that is between 15 and 30
seconds after the minute is the area under the uniform density
function.
Area
= P(15 < x < 30)
= 15/60
= 0.25
15
30
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Relative frequency histograms that are symmetric and
bell-shaped are said to have the shape of a normal
curve.
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If a continuous random variable is normally distributed,
or has a normal probability distribution, then a relative
frequency histogram of the random variable has the
shape of a normal curve (bell-shaped and symmetric).
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EXAMPLE
A Normal Random Variable
The data on the next slide represent the heights
(in inches) of a random sample of 50 two-year
old males.
(a) Draw a histogram of the data using a lower
class limit of the first class equal to 31.5 and a
class width of 1.
(b) Do you think that the variable “height of 2year old males” is normally distributed?
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36.0
34.7
34.4
33.2
35.1
38.3
37.2
36.2
33.4
35.7
36.1
35.2
33.6
39.3
34.8
37.4
37.9
35.2
34.4
39.8
36.0
38.2
39.3
35.6
36.7
37.0
34.6
31.5
34.0
33.0
36.0
37.2
38.4
37.7
36.9
36.8
36.0
34.8
35.4
36.9
35.1
33.5
35.7
35.7
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36.8
34.0
37.0
35.0
35.7
38.9
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In the next slide, we have a normal density
curve drawn over the histogram. How does
the area of the rectangle corresponding to a
height between 34.5 and 35.5 inches relate
to the area under the curve between these
two heights?
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EXAMPLE
Interpreting the Area Under a Normal Curve
The weights of giraffes are approximately normally distributed with mean μ =
2200 pounds and standard deviation σ = 200 pounds.
(a) Draw a normal curve with the parameters labeled.
(b) Shade the area under the normal curve to the left of x = 2100 pounds.
(c) Suppose that the area under the normal curve to the left of x = 2100
pounds is 0.3085. Provide two interpretations of this result.
(a), (b)
(c)
• The proportion of giraffes whose
weight is less than 2100 pounds is
0.3085
• The probability that a randomly
selected giraffe weighs less than
2100 pounds is 0.3085.
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EXAMPLE
Relation Between a Normal Random Variable and a
Standard Normal Random Variable
The weights of giraffes are approximately normally distributed with mean μ =
2200 pounds and standard deviation σ = 200 pounds. Draw a graph that
demonstrates the area under the normal curve between 2000 and 2300
pounds is equal to the area under the standard normal curve between the Zscores of 2000 and 2300 pounds.
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