Chapter 7 The Normal Probability Distribution

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Chapter 7The Normal Probability Distribution

• 7.1
• Properties of the Normal Distribution

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Objectives

• Understand the uniform probability distribution
• Graph the normal density curve
• State the properties of the normal curve
• Understand the role of area in the normal density
  function
• Understand the relation between a normal random
  variable and a standard normal random variable
  with regard to area.

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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
  from10 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.

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• EXAMPLE Illustrating the Uniform Distribution
• That is to say your package is just as likely to
  arrive between 1015 and 1016 as it is to arrive
  between 1040 and 1041.
• The random variable X can be any value in the
  interval from 0 to 60, that is, 0 lt X lt 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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• EXAMPLE Illustrating the Uniform Distribution
• For continuous random variables, we cant just
  substitute the value of the random variable into
  a probability distribution function since the
  probability of observing any one specific value
  is zero.
• For example the probability that the package
  arrives exactly 3.4534 minutes after 10 is zero
  since the classical probability is the number of
  ways an event can occur over the total number of
  possible outcome.
• The number of possible outcomes is infinite.

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Probability Density Function A probability
density function is an equation that is used to
compute probabilities of continuous random
variables that must satisfy the following two
properties. 1. The area under the graph of the
equation over all possible values of the random
variable must equal one. 2. The graph of the
equation must be greater than or equal to zero
for all possible values of the random variable.
That is, the graph of the equation must lie on or
above the horizontal axis for all possible values
of the random variable.
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Area as Probability
The area under the graph of a density function
over some interval represents the probability of
observing a value of the random variable in that
interval.
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EXAMPLE Area as a Probability Referring to the
earlier example, what is the probability that
your package arrives between 1010 am and 1020
am? The green rectangle has an area of
101/600.167 Therefore the probability is 16.7
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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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Normal Distribution
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). The mean
shifts the curve left or right. The standard
deviation changes how wide or narrow the bell
curve is.
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• Properties of the Normal Density Curve
• The Empirical Rule
• about 68 of the area under the graph is within
  one standard deviation of the mean
• about 95 of the area under the graph is within
  two standard deviations of the mean
• about 99.7 of the area under the graph is
  within three standard deviations of the mean.

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EXAMPLE A Normal Random Variable The following
data represent the heights (in inches) of a
random sample of 50 two-year old males. (a)
Create a relative frequency distribution with the
lower class limit of the first class equal to
31.5 and a class width of 1. (b) Draw a histogram
of the data. (c ) Do you think that the variable
height of 2-year old males is normally
distributed?
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36.0 36.2 34.8 36.0 34.6 38.4 35.4 36.8 34.7 33.4
37.4 38.2 31.5 37.7 36.9 34.0 34.4 35.7 37.9 39.3
34.0 36.9 35.1 37.0 33.2 36.1 35.2 35.6 33.0 36.8
33.5 35.0 35.1 35.2 34.4 36.7 36.0 36.0 35.7 35.7
38.3 33.6 39.8 37.0 37.2 34.8 35.7 38.9 37.2 39.3
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In the next slide, we have a normal density curve
drawn over the histogram. It has a mean of 36 or
µ 36.
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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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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? The area of the rectangle represents the
proportion of males between 34.5 and 35.5 or the
probability of randomly selecting a male from the
population that is between 34.5 and 35.5 The area
is 1 0.20 0.20
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EXAMPLE Interpreting the Area Under a Normal
Curve The weights of pennies minted after 1982
are approximately normally distributed with mean
2.46 grams and standard deviation 0.02 grams. If
you draw a normal curve with the parameters
labeled and shade the region under the normal
curve between 2.44 and 2.49 grams. If the area
under the curve for the shaded region is
0.7745.Provide two interpretations for this area.
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EXAMPLE Interpreting the Area Under a Normal
Curve The weights of pennies minted after 1982
are approximately normally distributed with mean
2.46 grams and standard deviation 0.02 grams. If
you draw a normal curve with the parameters
labeled and shade the region under the normal
curve between 2.44 and 2.49 grams. If the area
under the curve for the shaded region is
0.7745.Provide two interpretations for this
area. The proportion of pennies minted after 1982
is .7745 or the probability that a random
selected penny was minted after 1982 is 77.45
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EXAMPLE Relation Between a Normal Random
Variable and a Standard Normal Random
Variable The heights of pediatricians 200
three-year-old female patients have a normal
distribution with a mean of 38.72 and a standard
deviation of 3.17. Draw a graph that demonstrates
the area under the normal curve between 35 and 38
inches tall is equal to the area under the
standard normal curve between the Z-scores of 35
and 38 inches tall.
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EXAMPLE Relation Between a Normal Random
Variable and a Standard Normal Random
Variable We find the z-scores as follows Z X
µ 35 38.72 -1.17 s 3.17 Z X µ 38
38.72 -0.23 s 3.17
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EXAMPLE Relation Between a Normal Random
Variable and a Standard Normal Random Variable
The graphs demonstrate that the area under the
normal curve between 35 and 38 inches tall is
equal to the area under the standard normal curve
between the Z-scores of -1.17 and -0.23.