Question

2a. According to a Pew Research survey, about 25% of American adults are pessimistic about the future of marriage and the family. This is based on a sample, but assume this percentage is correct for all American adults. What is the probability that in a sample of 20 American adults, 23% or fewer of the people in the sample are pessimistic about the future of marriage and family

Answer 1

Answer: 0.4181807

Explanation:

Probability that American adults are pessimistic about the future of marriage and the family: p= 0.25

Sample size n= 20

Let
`\hat{p}` be the sample proportion of American adults that are pessimistic about the future of marriage and the family.

`P(\hat{p}\leq0.23)=P(\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}<\frac{0.23-0.25}{\sqrt{(0.25(1-0.25))/(20)}})\\\\= P(Z<-0.20655) \ \ \ \ [Z=\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}]\\\\=1-P(Z<0.20655)\\\\= 1- 0.5818193=0.4181807`

Hence, the required probability = 0.4181807