Download Normal Approximation

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Normal Approximation
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Suppose we perform a sequence of n binomial
trials with probability of success p and
probability of failure q = 1 - p and observe the
number of successes. Then the histogram for the
resulting probability distribution may be
approximated by the normal curve with  = np
and   npq .

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
Binomial distribution with n = 40 and p = .3
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A plumbing-supplies manufacturer produces
faucet washers that are packaged in boxes of
300. Quality control studies have shown that 2%
of the washers are defective. What is the
probability that more than 10 of the washers in a
single box are defective?
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Let X = the number of defective washers in a box.
X is a binomial random variable with
n = 300 and p = .02.
We will use the approximating normal curve with
 = 300(.02) = 6 and   300 .02.98  2.425.
Since the right boundary of the X = 10 rectangle
is 10.5, we are looking for Pr(X > 10.5).
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10.5  6
z
 1.85
2.425
The area of the region to the right of 1.85 is
1 - A(1.85) = 1 - .9678 = .0322.
Therefore, 3.22% of the boxes should contain
more than 10 defective washers.
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Probabilities associated with a binomial
random variable with parameters n and p can
be approximated with a normal curve having
 = np and

  np 1  p .
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