Question

Thirty percent of the registered voters in a state are women. Let W be the number of women in a random sample of 500 registered voters.

(a) Find the exact probability P(140 ≤ W ≤ 160).

(b) Use the normal approximation to the binomial distribution given by the CLT to compute the probability P(140 ≤ W ≤ 160).

(c) Since the number of women is necessarily integer, some authorities would suggest that, when approximating the probability of Part (a) using the CLT, one should compute P(139.5 ≤ W ≤ 160.5) in Part (b). Is this closer to the exact probability?

Answer 1

(a) W follows a binomial distribution with p = 0.30 and n = 500

q = 1 - p = 0.70

P(140 ≤ W ≤ 160) = BINOMDIST(160, 500, 0.3, TRUE) - BINOMDIST(139, 500, 0.3, TRUE) [Using Excel]

= 0.84719 - 0.15269 = 0.6945

(b) μ = np = 500 * 0.3 = 150, σ = √(npq) = √(500 * 0.3 * 0.7) = 10.247

z = (x - μ)/σ

z1 = (140 - 150)/10.247 = -0.9759 and z2 = (160 - 150)/10.247 = 0.9759

P(140 ≤ W ≤ 160) = P(-0.9759 < z < 0.9759) = 0.6709

(c) z1 = (139.5 - 150)/10.247 = -1.0247 and z2 = (160.5 - 150)/10.247 = 1.0247

P(140 ≤ W ≤ 160) = P(-1.0247 < z < 1.0247) = 0.6945

Yes, this result is same as the exact probability obtained in (a)

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