Answer: 0.4181807
Step-by-step explanation:
Probability that American adults are pessimistic about the future of marriage and the family: p= 0.25
Sample size n= 20
Let [tex]\hat{p}[/tex] be the sample proportion of American adults that are pessimistic about the future of marriage and the family.
[tex]P(\hat{p}\leq0.23)=P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}<\dfrac{0.23-0.25}{\sqrt{\dfrac{0.25(1-0.25)}{20}}})\\\\= P(Z<-0.20655) \ \ \ \ [Z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}]\\\\=1-P(Z<0.20655)\\\\= 1- 0.5818193=0.4181807[/tex]
Hence, the required probability = 0.4181807
