Answer
Confidence interval: $0.317\lt p ̂\lt0.383$
Work Step by Step
$p̂ =\frac{x}{n}=\frac{275}{785}=0.350$
Required condition:
$np̂ (1-p̂ )=785\times0.350(1-0.350)=178.59\gt10$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$z_{\frac{α}{2}}=z_{0.025}$
If the area of the standard normal curve to the right of $z_{0.025}$ is 0.025, then the area of the standard normal curve to the left of $z_{0.025}$ is $1−0.025=0.975$
According to Table V, the z-score which gives the closest value to 0.975 is 1.96.
$Lower~bound=p ̂-z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n}}=0.350-1.96\times\sqrt {\frac{0.350(1-0.350)}{785}}=0.317$
$Upper~bound=p ̂+z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n}}=0.350+1.96\times\sqrt {\frac{0.350(1-0.350)}{785}}=0.383$
