normal approximation with continuity correction

Question

n =30 and p=.6, find P (x≤ 15) (use normal approximation with continuity correction)

Answer 1

If n = 30, p is constant and = .6, outcomes are dichotomous (i.e. that the distribution is binomial) then the following hold true.

mean = np = 30(.6) = 18. variance = (30)(.6)(.4))= 7.2 {npq where q=1-p}. sd = sqrt(7.2) = 2.683

note: continuity correction was requested.

as defined the continuity correction assumes that P(X≤ x) = P(X<x+1) and

for any x ∈ {0, 1, 2, ... n}. If np and np(1 − p) are large (sometimes taken as both ≥ 5), then the probability above is fairly well approximated by

P(Y≤ x+1/2)

calculating the Z score with the adjustment. Z = (15.5-18)/2.683 = -.9318 i.e. ((x₁+.5)-x₀)/σ

P(Y≤15.5) = = .205

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note: using binomial distribution instead of the normal approximation for the exact answer

. TI-8x bicdf(30,6,15) = .1754) which is not far off under the circumstances.