PROBLEM: Suppose X is Bi(n,p). I would like to know P(X<a). How do I use the Normal Approximation to estimate this probability?
NOTE: In the old and golden days, there were no computers. Then it was VERY difficult to do this calculation using the Binomial Probabilities. Calculating the approximation was MUCH easier, as it was possible to do it using a TABLE of normal probabilities. NOWADAYS, this problem is not very important, because our computers can directly compute exactly the Binomial probabilities. The required probability is =BINOMDIST(a,n,p,1). There is no need to approximate, However the concept of Normal approximation remains important for many purposes.
SOLUTION via CLT: Let M=np be the mean of X. Let S=sqrt(n*p*(1-p)) be the standard error of X. Let Z=(X-M)/S. According to the CLT, Z should be approximately standard normal. We can use this to calculate the required probability:
P(X<a)=P( [X-M]/S < (a-M)/S). = P( Z < (a-M)/S ) = NORMSDIST( (a-M)/S ).
The EXCEL Spreadsheet BinomialNormal assesses the accuracy of this approximation over a suitable range of values of a. We describe the entries on the spreadsheet, attached below
FIRST ROW: B1: p, D1: n F1: M=np G1: SE=sqrt(np(1-p), Skew=(1-2p)/SE Kurt = (1 - 6p(1-p))/SE^2
COL A: Percentiles of the Normal distribution: 0.01, 0.05,0.10,0.15,...,0.95,0.99 -- checking fit at these percentiles.
COL B: =NORMSINV(Ai): This gives the Z-value at which NormDist achieves this percentile.
COL C: M+Z*SE Converts Z-value in Col B to equivalent for Binomial -- this is Z Std Errors away from Mean M.
COL D: Floor (C1) rounds value in Col C to the nearest integer Below
COL E: BINOMDIST(D1, N, P, true) gives the binomial probability for value in col D.
COL F: NORMSDIST((D1-M)/SE) give the normal approximation to the Binomial Probability
COL G: Calculates Absolute Error of Approximation: Difference between Normal and Binomial
SECOND ROW: G2 Max Error of Approximation I2: Max Error AFTER Continuity Correction
COL H: NORMDIST((0.5+D1-M)/SE) give normal approximation AFTER Continuity Correction
EXERCISE: Use the EXCEL Spreadsheet to create a TABLE of values for SKEW, KURT and Maximum Error of Approximation. Vary Values of N,P to get different values of Skewness and Kurtosis. ASSESS the sensitivity of the Maximum Error to Skewness and Kurtosis. EVALUATE the rules for WHICH values of N, P, 1-P lead to a suitable approximation of Binomial by Normal