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Question 1176059: Show and explain what is the normal distribution approximation for the binomial distribution where n = 20 and p = .25 (i.e. the binomial distribution displayed in Figure 1 of Binomial Distribution)?
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! n*p is 5 and n(1-p) is 15. Ten is usually considered the minimal number, but one can do the approximation and compare it.
mean is np=5
variance is np(1-p)=3.73
sd is sqrt (V)=1.94
compare the probability of getting 4 or fewer,
exact probability is 0.4148
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normal approximation uses mean of 5 and 4.5 rather than 4 to deal with the difference between a mass function (binomial) and a density function (normal)
z< =(4.5-5)/1.94
or -0.5/1.94
<-0.26
that probability is 0.3974
Still not a bad approximation considering the conditions were not met.
AS the binomial test has a large np and n(1-p), it is more and more similar to the normal curve with np as mean and sqrt(np(1-p)) for sd.
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