Let p be the population proportion of adult Americans could name at least one justice who is currently serving on the U.S. Supreme Court.
As per given , we have
[tex]H_0: p\geq0.5\\\\ H_a:p<0.5[/tex] , since alternative hypothesis is left-tailed , so the test is left-tailed.
Sample size : n= 1000
Sample proportion : [tex]\hat{p}=\dfrac{390}{1000}=0.39[/tex]
Test statistic : [tex]t=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
[tex]t=\dfrac{0.39-0.5}{\sqrt{\dfrac{0.5(1-0.5)}{1000}}}=-6.96[/tex]
P-value : [tex]P(z<-6.96)=1-P(z<6.96)=1-0.9999=0.0001[/tex] [using p-value table for z.]
Decision : Since P-value < Significance level , so we reject the null hypothesis.
Conclusion : We have enough evidence to support the claim that fewer than half of adult Americans can name at least one justice currently serving on the Supreme Court.
