Download Stat211 Labpack/Normal Approximation of Binomial

Stat211 Labpack/Normal Approximation of Binomial
Normal Approximation of Binomial Probabilities:
Goals of the lab.
• To gain an understanding of how the Normal distribution can be used to approximate the Binomial
distribution.
Background:
The binomial distribution begins to take on a shape like the Normal distribution, when n is large and p is
close to 0.5. Specifically, the conditions for the Normal distribution to approximate the Binomial are that
np>5 and n(1-p) >5. Since the Normal distribution is a continuous distribution and the Binomial is a
discrete distribution, we must make corrections (called the continuity correction) to the probabilities
calculated from the Normal distribution to account for this. For this lab, we will use X is a binomial RV
and Y is a Normal RV used to approximate X. Y is chosen to have the same mean and standard deviation
as X, m=np and s=(np(1-p))0.5, respectively.
Relevant Formulae:
Below are the corrections for continuity for the Normal approximation to the Binomial.
Desired Probability
P(X>r)
Calculated Probability
P(Y>r+0.5)
P(X≥r)
P(Y>r-0.5)
P(X<r)
P(Y<r-0.5)
P(X£r)
P(Y< r+ 0.5)
P(X=r)
P(Y<r+0.5) – P(Y<r-0.5)
Example:
Suppose that X is a binomial RV with n=80 and p = 0.4. Since np=80*.4 = 32>5 and n(1-p) = 80*0.6 =
48>5, then we can create a Normal RV to approximate X. The mean of this RV, call it Y, is m=np=32 and
the standard deviation is s=4.382
Find P(X>35)
If we want P(X>35), we use P(Y ≥ 36) = P(Y>35.5) = P(Z>(35.5-32)/4.832) = P(Z>0.80) = 1-P(Z<0.80) =
1-0.7881 = 0.2119.
Find P(X£30)
P(X£30) is approximately = P(Y<30.5) = P(Z<-0.34) = 0.3669
Find P(X≥40) is approximately = P(Y>39.5) = P(Z>1.71) = 1-P(Z<1.71) = 1 - 0.9564 = 0.0436
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