The 90% confidence interval for the proportion of all Americans that believe who it is the government's responsibility is (0.117, 0.145).
For a proportion of [tex]\pi[/tex] in a sample of size n, with a confidence level of [tex]\alpha[/tex], the confidence interval for the proportion is as follows:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the z-score that has a p-value of [tex]\frac{1+\alpha}{2}[/tex].
For this question, 206 people said that it definitely should out of 1572 randomly selected people.
This means that [tex]n = 1572, \pi = \frac{206}{1572} = 0.1310[/tex]
90% confidence level
So [tex]\alpha = 0.9[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.9}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.131 - 1.645\sqrt{\frac{0.131(0.869)}{1572}} = 0.117[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.131 + 1.645\sqrt{\frac{0.131(0.869)}{1572}} = 0.145[/tex]
The confidence interval is (0.117, 0.145).
A similar problem is given at https://brainly.com/question/16807970
