Q:

In a representative sample of 1000 adult Americans, only 390 could name at least one justice who is currently serving on the U.S. Supreme Court. Using a significance level of 0.01, carry out a hypothesis test to determine if there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one justice currently serving on the Supreme Court. (Round your test statistic to two decimal places and your P-value to four decimal places.)

Accepted Solution

A:
Answer with explanation: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= 1000Sample 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.