A researcher studying public opinion of proposed Social Security changes and asks a random sample of adult Americans whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, how many adult Americans does the researcher need to sample in the following cases (a) if 37% of all adult Americans support the changes?
(b) 25% of all adults Americans support the changes?

Complete question is;

A researcher studying public opinion of proposed Social Security changes and asks a random sample of 30 adult Americans whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, how many adult Americans does the researcher need to sample in the following cases

(a) if 37% of all adult Americans support the changes?

(b) 25% of all adults Americans support the changes?

A) 13 adults

B) 23 adults

Step-by-step explanation:

A) We are given;

p = 37% = 0.37

Since we are told that this is an approximately normal distribution, we will use the formula;

np(1 - p) ≥ 10

Thus;

0.37n(1 - 0.37) ≥ 10

n ≥ 10/(0.63 × 0.37)

n ≥ 42.9

We need a whole number, thus n = 43

From the question, we have already 30 adults , so the required number of adults are; 43 - 30 = 13 adults

B) We are given;

p = 25% = 0.25

Since we are told that this is an approximately normal distribution, we will use the formula;

np(1 - p) ≥ 10

Thus;

0.25n(1 - 0.25) ≥ 10

n ≥ 10/(0.75 × 0.25)

n ≥ 53.33

n ≈ 53

From the question, we have already 30 adults , so the required number of adults are; 53 - 30 = 23 adults

i believe the correct answer is b.