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 = 200?(Round your answer to four decimal places.)


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.)

Answer 1

1)

for normal distribution z score =(p̂-p)/σp
here population proportion=     p=	0.490
sample size       =n=	200
std error of proportion=σp=√(p*(1-p)/n)=	0.0353

probability =

P(0.46<X<0.52)

=

P(-0.85<Z<0.85)=

0.8023-0.1977=

0.6046

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