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Chapter 9: The Normal Probability Distribution
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Recall, in order for us to use the normal distribution to find probabilities we
will need to know the mean and standard deviation of the normal random
variable. Since we are approximating the binomial distribution with a
normal distribution we will use the mean and the standard deviation of the
binomial distribution in the approximation. These are given below.
Example 9-13
:
During pre-registration of incoming freshmen at a large
university, 25% of them choose a major in the College of Business. Find the
probability that in a random sample of 1000 incoming freshmen, at least 275
of them opted for a major in the College of Business.
Solution:
Let
X
= number of incoming freshmen in the sample of 1000
who will choose a major in the College of Business.
X
will be a binomial random variable since we have a fixed number of trials
(
= 1000); two possible outcomes (either a freshman chooses a major in
business or not); the probability that an incoming freshman will major in a
Business program is fixed (
= 0.25); an incoming freshman will choose a
business major independent of other incoming freshmen.
Next we need to check whether the criterion for the normal approximation to
the binomial distribution holds. That is, we need to check whether
> 5
and
–
> 5. Now,
= 1000
0.25 = 250 and
–
= 1000
0.75
= 750. Since both
> 5 and
–
> 5 then we can use the normal
approximation to the binomial distribution. The mean
= 1000
0.25 = 250 and the standard deviation
√
=
√
= 13.6931.
Now we need to compute
P
(
X
275) =
P
(
X
275 – 0.5) =
P
(
X
274.5)
P
(
X
> 274.5).