Answer:
a) (68%,72%)
b) (67%,73%)
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 1482
Proportion of reported that they either sometimes or often think about social issues =
[tex]\hat{p} = 70\% = 0.7[/tex]
a) 95% Confidence interval:
[tex]\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
Putting the values, we get:
[tex]0.7\pm 1.96(\sqrt{\dfrac{0.7(1-0.7)}{1482}})\\\\ = 0.7\pm 0.023\\\\=(0.6767,0.7233)\approx (68\%,72\%)[/tex]
b) 99% Confidence interval:
[tex]\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
[tex]z_{critical}\text{ at}~\alpha_{0.01} = 2.58[/tex]
Putting the values, we get:
[tex]0.7\pm 2.58(\sqrt{\dfrac{0.7(1-0.7)}{1482}})\\\\ = 0.7\pm 0.0301\\\\=(0.6693,0.7307)\approx (67\%,73\%)[/tex]
