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

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 = 250? (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.)

Explanation / Answer

p = 0.49, n = 250
SE = sqrt(p*(1-p)/n) = sqrt(0.49*0.51/250) = 0.0316

P(0.46 < pcap < 0.52)
= P((0.46 - 0.49)/0.0316 < z < (0.52 - 0.49)/0.0316)
= P(-0.9494 < z < 0.9494)
= P(z < 0.9494) - P(z < -0.9494)
= 0.8288 - 0.1712
= 0.6576

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

P(0.46 < pcap < 0.52)
= P((0.46 - 0.49)/0.0071 < z < (0.52 - 0.49)/0.0071)
= P(-4.225 < z < 4.225)
= P(z < 4.225) - P(z < -4.225)
= 0.99998 - 0.000012
= 0.999976

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