Question

A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 25 adult Americans and asks them 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 more adult Americans does the researcher need to sample in the following​ cases? ​(a) 15​% of all adult Americans support the changes ​(b) 20​% of all adult Americans support the changes ​(a) The researcher must ask nothing more American adults. ​(Round up to the nearest​ integer.) ​(b) The researcher must ask nothing  more American adults. ​(Round up to the nearest​ integer.)

Answer 1

Here sample size = 25

here we say that the distribution of the sample proportion is approximatly normal.

So here to approximate binomial distribution to normal distribution. It requires that np(1-p) >= 10

(a) so here p = 0.15

n * 0.15 * (1 - 0.15) >= 10

n >= 10/(0.15 * 0.85)

n>= 78.43

or n = 79

we already got 25 american adults so we require 79 - 25 = 54 more American adults.

(b) Now here p = 0.20

n * 0.20 * (1 - 0.20) >= 10

n >= 10/(0.2 * 0.8)

n >= 62.5

so n= 63

we already got 25 American adults , so we require 63 - 25 = 38 more American adults.