\"What do you think is the ideal number of children for a family to have?\" A Ga
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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.)
Combine these results to make a general statement about the effect of larger samples in a sample survey. (Which is the correct statment?)
a)Larger samples have no effect on the probability that p will be close to the true proportion p.
b)Larger samples give a larger probability that p will be close to the true proportion p.
c)Larger samples give a smaller probability that p will be close to the true proportion p.
Explanation / Answer
a)
99.7% lies within 3 standard deviation from mean
P(0.46 < X < 0.52) = P(X < 0.52) - P( X< 0.46)
= P(Z < 0.52 - 0.49/0.0316) - P(Z < 0.46 - 0.49 /0.0316)
= P(Z < 0.9494) - P(z < -0.9494)
= 0.8288 - 0.1712
= 0.6576
b)
standard error = sqrt( 0.49 * 0.51 /5000) = 0.00707
probability = P(Z < 0.52 - 0.49/0.00701) - P(Z < 0.46 - 0.49 /0.00707)
= P(Z < 4.2796 ) - P(Z < -4.24)
= 1 - 0
= 1
a)Larger samples have no effect on the probability that p will be close to the true proportion p.
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