Answer with explanation:
Let p be the population proportion of Americans disapprove of the job Congress is doing .
As per given , we have
[tex]H_0:p=0.86\\\\ H_a:p<0.86[/tex]
Since [tex]H_a[/tex] is left-tailed , so the test is a left-tailed test.
Also, it is given that : In a poll of 1000 Americans during Summer 2013 by NBC News, 830 said they disapprove of the job Congress is doing.
i.e. n= 1000
[tex]\hat{p}=\dfrac{830}{1000}=0.83[/tex]
Test statistic : [tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
[tex]z=\dfrac{0.83-0.86}{\sqrt{\dfrac{0.86(1-0.86)}{1000}}}=-2.734[/tex]
P-value for left tailed test : P(z<-2.734)=1-P(z<2.734)
[∵ P(Z<-z)=1-P(Z<z)]
=1-0.99687=0.00313
Decision : Since p-value (0.00313) < significance level (0.01), so we reject the null hypothesis.
[Rejection criteria : if p-value is less than [tex]\alpha[/tex] (significance level ), then null hypothesis gets rejected.]
Thus , we conclude that we have enough evidence to support the claim at 0.01 significance level that the proportion of Americans who disapprove of the job Congress is doing has decreased.
