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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 Page 1 of 1