\"What do you think is the ideal number of children for a family to have?\" A Ga

Question

"What do you think is the ideal number of children for a family to have?" A Gallup Poll asked this question of 1016 randomly chosen adults. Almost half (49%) thought two children was ideal.† We are supposing that the proportion of all adults who think that two children is ideal is p = 0.49.
What is the probability that a sample proportion p falls between 0.46 and 0.52 (that is, within ±3 percentage points of the true p) if the sample is an SRS of size n = 200? (Round your answer to four decimal places.)


What is the probability that a sample proportion p falls between 0.46 and 0.52 if the sample is an SRS of size n = 5000? (Round your answer to four decimal places.)


Combine these results to make a general statement about the effect of larger samples in a sample survey.

Larger samples have no effect on the probability that p will be close to the true proportion p. Larger samples give a smaller probability that p will be close to the true proportion p.     Larger samples give a larger probability that p will be close to the true proportion p.

Explanation / Answer

(a) = 0.49, SE = ((1 - )/n) = ((0.49 * 0.51)/200) = 0.0353

z1 = (p1 - )/SE = (0.46 - 0.49)/0.0353 = -0.8487 and z2 = (p2 - )/SE = (0.52 - 0.49)/0.0353 = 0.8487

P(0.46 < p < 0.52) = P(-0.8487 < z < 0.8487) = 0.6040

(b) = 0.49, SE = ((1 - )/n) = ((0.49 * 0.51)/5000) = 0.0071

z1 = (p1 - )/SE = (0.46 - 0.49)/0.0071 = -4.2435 and z2 = (p2 - )/SE = (0.52 - 0.49)/0.0071= 4.2435

P(0.46 < p < 0.52) = P(-4.2435 < z < 4.2435) = 1.0000

(c) Larger samples give a larger probability that p will be close to the true proportion p.

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