S3 De-Moivre Laplace CLT

This theorem says that we can estimate Binomial Probabilities by Normal Probabilities. More specifically, Let S be Bi(n,p). Then ES=np and Var(S)=np(1-p). In notation used above, q=1-p and so Var(S)=npq. Define Z to be the standardization of S:

Z=(S-n)/Sqrt(np(1-p)). Then the Central Limit Theorem says that Z is approximately standard Normal for large values of n. The above theorem looks at P(a<Z<b) and says that this can be approximated by standard Normal probabilities. We now translate this theorem into the language of EXCEL commands. If Y is standard Normal, then P(Y<b)= NORMSDIST(b), and P(Y<a)= NORMSDIST(a). Therefore, P(a<Y<b)=NORMSDIST(b)-NORMSDIST(a). Let SE be the standard error of S, so that

SE=SQRT(n*p*(1-p)), P(Z<b)=P((S-np)/SE < b) = P(S< n*p + b*SE) = BINOMDIST(n*p+b*SE,n,p,1). Similarly, P(Z<a) =

BINOMDIST(n*p+a*SE,n,p,1). So P(a<Z<b)=BINOMDIST(n*p+b*SE,n,p,1)-BINOMDIST(n*p+a*SE,n,p,1). According to the DeMoivre Laplace theorem, these two probabilites should be close to each other for large n.