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 = 300? (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.)


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

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

Answer 1

Answer:

Let p be the proportion of all adults who think that two children is ideal

p = 0.49

n = 350

And we have to find P(0.46 < p^ < 0.52)

Here we use sampling distribution of proportion.

It says that the mean of the distribution of sample proportions is equal to the population proportion (p). The standard deviation of the distribution of sample proportions is symbolized by SE(ˆp) and equals √p(1−p)n; this is known as the standard error of ˆp.

Mean = p = 0.49

q = 1 - p = 1 - 0.49 = 0.51

SE = sqrt((0.49*0.51)/300) = 0.0286

Now convert p^ = 0.46 and p^ = 0.52 into z-score.

z-score is defined as,

z = (p^ - p) / SE

z = (0.46 - 0.49) / 0.0286 = -1.05

z = (0.52 - 0.49) / 0.0286 = 1.05

P(-1.05 Z < 1.05) = P(Z < 1.05) - P(Z < -1.05)

                                      = 0.8531 – 0.1469 = 0.7062

Similarly, For n = 5000,

SE = sqrt(0.49*0.51)/5000 = 0.007

z = (0.46 - 0.49) / 0.007 = -4.29

z = (0.52 - 0.49) / 0.007 = 4.29

P(-4.29 Z < 4.29) = P(Z < 4.29) - P(Z < -4.29)

                            = 1 – 0 = 1

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