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 p̂ 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 = 400? (Round your answer to four decimal places.)

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What is the probability that a sample proportion p̂ falls between 0.46 and 0.52 if the sample is an SRS of size n = 5000? (Round your answer to four decimal places.)
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Combine these results to make a general statement about the effect of larger samples in a sample survey. choose one ?

A.Larger samples have no effect on the probability that p̂ will be close to the true proportion p.

B.Larger samples give a larger probability that p̂ will be close to the true proportion p.   

C.Larger samples give a smaller probability that p̂ will be close to the true proportion p.

Answer 1

1)

here population proportion=     p=	0.490
sample size       =n=	400
std error of proportion=σp=√(p*(1-p)/n)=	0.0250

probability that a sample proportion p̂ falls between 0.46 and 0.52 :

probability =

P(0.46<X<0.52)

=

P(-1.2<Z<1.2)=

0.8849-0.1151=

0.7698

2)

sample size       =n=	5000
std error of proportion=σp=√(p*(1-p)/n)=	0.0071

probability =

P(0.46<X<0.52)

=

P(-4.24<Z<4.24)=

1-0=

1.0000

3)

B.Larger samples give a larger probability that p̂ will be close to the true proportion p.