If X is a binomial r.v. with parameters n and p. Already showed if n is large (n

Question

If X is a binomial r.v. with parameters n and p. Already showed if n is large (n > 20) and p is small (p 5 and n(1 - p) > 5, then the Normal approximates the Binomial. Ex: Suppose that 40% of the people in the US population are blood type O positive. If you randomly sample 20 individuals, how many would we expect to be blood type O+? Binomial Distribution: Normal Distribution: What is the probability of observing less than 10 individuals, in the sample of 20, with blood type O+? Binomial: Standard Normal:

Explanation / Answer

p = .40
n = 20

Binomial dist:E(X) = expect to be blood type O+ = np = .40*20 = 8
Normal dist: E(X) = Since np=8>5 and n(1-p) = .6*20 = 12>5, we have normal approx of binomial

Binmoial:P(X<10) = 20C0(.4^20)(.6^0)+..+20C10(.4^10)(.6^10) = 0.872
Normal: P(X<10) = (10-np)/sqrt(npq) = (10-8)/sqrt(3.2) = P(Z<1.118) = .8686

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