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Chapter 9: The Normal Probability Distribution
401
In this section we will discuss this normality approximation to the binomial
distribution and use this approximation to find probabilities.
Based on these observations in
Section 9-2
, statisticians have come up with
several criteria to help determine when approximating the binomial
probability distribution with the normal probability distribution is
appropriate. Following are some of them.
•
The normal approximation is appropriate when
> 5 and
–
> 5 where
is the number of trials in a binomial
experiment and
is the fixed probability of a successful outcome
in the experiment.
•
The normal approximation is appropriate when
5 and
–
5
•
The normal approximation is appropriate when
10 and
–
10.
•
The normal approximation is appropriate if
0
± 3
√
.
For this e-book, we will use the criterion for the normal approximation to the
binomial distribution of
> 5 and
–
> 5. Since the normal
distribution is continuous, in order for us to use it to approximate the
discrete binomial distribution, we would have to apply a correction factor
called the
continuity correction factor
. A continuity correction factor is a
correction which is applied when a continuous distribution is used to
approximate a discrete distribution.
The correction is to either add or
subtract 0.5 of a unit from each discrete x-value.
Next we will present the different possibilities of applying the correction
factor.
Case 1:
P
(X =
a
) where
a
is the value of a Binomial random
variable X
For instance, if we needed to find
P
(
X
= 50), when
X
is a binomial random
variable then we will have to apply the continuity correction factor in order
to use the normal approximation since
X
is discrete. In this particular case
we will have to extend to the interval of 49.5 to 50.5.