Answer
We are 95% confident that the population proportion of Americans who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage falls between 0.440 and 0.500.
Work Step by Step
$x$ = 493, n = 1050, $\hat{p}$ = 0.470, 95% confidence
i) Find $\frac{\alpha}{2}$
$\alpha = 1 - 0.95$ = 0.05
$\frac{0.05}{2}$ = 0.025
ii) The z-score that corresponds to 0.025 is 1.96
iii) Find margin of error:
E = $z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = 1.96\cdot \sqrt{\frac{0.470(0.53)}{1050}}$ $\approx$ 0.03019
iv) Find lower bound of CI:
$\hat{p}$ - E
= 0.470 - 0.03019
= 0.43981
$\approx$ 0.440
v) Find upper bound of CI:
$\hat{p}$ + E
= 0.470 + 0.03019
= 0.50019
$\approx$ 0.500
