8
PROBABILITY
DISTRIBUTIONS
AND STATISTICS
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8.6
Applications of the Normal
Distribution
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Approximating Binomial Distributions
3
Approximating Binomial Distributions
One important application of the normal distribution is that it
provides us with an accurate approximation of other
continuous probability distributions.
Here, we show how a binomial distribution may be
approximated by a suitable normal distribution.
This technique leads to a convenient and simple solution to
certain problems involving binomial probabilities.
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Approximating Binomial Distributions
We know that a binomial distribution is a probability
distribution of the form
P(X = x) = C(n, x)pxqn–x
x = 0, 1, 2, . . . , n
(19)
For small values of n, the arithmetic computations of the
binomial probabilities may be done with relative ease.
However, if n is large, then the work involved becomes
prodigious, even when tables of P(X = x) are available.
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Approximating Binomial Distributions
For example, if n = 50, p = .3, and q = .7, then the
probability of ten or more successes is given by
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Approximating Binomial Distributions
To see how the normal distribution helps us in such
situations, let’s consider a coin-tossing experiment.
Suppose a fair coin is tossed 20 times and we wish to
compute the probability of obtaining 10 or more heads.
The solution to this problem may be obtained, of course, by
computing
P(X  10) = P(X = 10) + P(X = 11) + · · · + P(X = 20)
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Approximating Binomial Distributions
The inconvenience of this approach for solving the problem
at hand has already been pointed out.
As an alternative solution, let’s begin by interpreting the
solution in terms of finding the area of suitable rectangles
of the histogram for the distribution associated with the
problem.
We may use Equation (19) to compute the probability of
obtaining exactly x heads in 20 coin tosses.
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Approximating Binomial Distributions
The results lead to the binomial distribution displayed in
Table 13.
Table 13
9
Approximating Binomial Distributions
Using the data from the table, we next construct the
histogram for the distribution (Figure 30).
Histogram showing the probability of obtaining x heads in 20 coin tosses
Figure 30
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Approximating Binomial Distributions
The probability of obtaining 10 or more heads in 20 coin
tosses is equal to the sum of the areas of the shaded
rectangles of the histogram of the binomial distribution
shown in Figure 31.
The shaded area gives the probability of obtaining 10 or more heads in 20 coin tosses.
Figure 31
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Approximating Binomial Distributions
Next, observe that the shape of the histogram suggests
that the binomial distribution under consideration may be
approximated by a suitable normal distribution.
Since the mean and standard deviation of the binomial
distribution are given by
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Approximating Binomial Distributions
respectively, the natural choice of a normal curve for this
purpose is one with a mean of 10 and standard deviation
of 2.24.
13
Approximating Binomial Distributions
Figure 32 shows such a normal curve superimposed on the
histogram of the binomial distribution.
The good fit suggests that
the sum of the areas of the
rectangles representing
P(X  10), the probability of
obtaining 10 or more heads
in 20 coin tosses, may be
approximated by the area of
an appropriate region under
the normal curve.
Normal curve superimposed on the
histogram for a binomial distribution
Figure 32
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Approximating Binomial Distributions
To determine this region, let’s note that the base of the
portion of the histogram representing the required
probability extends from x = 9.5 on, since the base of the
leftmost rectangle in the shaded region is centered at
x = 10 and the base of each rectangle has length 1
(Figure 33).
P(X 10) is approximated by the area under the normal curve.
Figure 33
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Approximating Binomial Distributions
Therefore, the required region under the normal curve
should also have x  9.5. Letting Y denote the continuous
normal variable, we obtain
P(X  10)  P(Y  9.5)
= P(Y > 9.5)
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Approximating Binomial Distributions
 P(Z > –0.22)
= P(Z < 0.22)
= .5871
Use the table of values of Z.
The exact value of P(X  10) may be found by computing
P(X = 10) + P(X = 11) + · · · + P(X = 20) in the usual
fashion and is equal to .5881.
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Approximating Binomial Distributions
Thus, the normal distribution with suitably chosen mean
and standard deviation does provide us with a good
approximation of the binomial distribution.
In the general case, the following result, which is a special
case of the central limit theorem, guarantees the accuracy
of the approximation of a binomial distribution by a normal
distribution under certain conditions.
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Approximating Binomial Distributions
Note:
It can be shown that if both np and nq are greater than 5,
then the error resulting from this approximation is
negligible.
19
Applications Involving
Binomial Random Variables
20
Applied Example 4 – Quality Control
An automobile manufacturer receives the microprocessors
that are used to regulate fuel consumption in its
automobiles in shipments of 1000 each from a certain
supplier.
It has been estimated that, on the average, 1% of the
microprocessors manufactured by the supplier are
defective.
Determine the probability that more than 20 of the
microprocessors in a single shipment are defective.
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Applied Example 4 – Solution
Let X denote the number of defective microprocessors in a
single shipment. Then X has a binomial distribution with
n = 1000, p = .01, and q = .99, so
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Applied Example 4 – Solution
cont’d
Approximating the binomial distribution by a normal
distribution with a mean of 10 and a standard deviation of
3.15, we find that the probability that more than 20
microprocessors in a shipment are defective is given by
Where Y denotes the normal
random variable
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Applied Example 4 – Solution
cont’d
 P(Z > 3.33)
 P(Z < –3.33)
= .0004
In other words, approximately 0.04% of the shipments
containing 1000 microprocessors each will contain more
than 20 defective units.
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Practice
p. 489 Self-Check Exercises #2
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