Answer
Confidence interval: $0.205\lt p ̂\lt0.307$
We are 90% confident that the proportion of adult Americans who would be willing to pay higher taxes if the revenue went directly toward deficit reduction is between 0.205 and 0.307.
Work Step by Step
$p̂=\frac{x}{n}=\frac{51}{199}=0.256$
Required condition:
$np̂ (1-p̂ )=199\times0.256(1-0.256)=37.9\gt10$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.90=1-α$
$α=0.1$
$z_{\frac{α}{2}}=z_{0.05}$
If the area of the standard normal curve to the right of $z_{0.05}$ is 0.05, then the area of the standard normal curve to the left of $z_{0.05}$ is $1−0.05=0.95$
According to Table V, there are 2 z-scores which give the closest value to 0.95: 1.64 and 1.65. So, let's find the mean of these z-scores: $\frac{1.64+1.65}{2}=1.645$
$Lower~bound=p ̂-z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n}}=0.256-1.645\times\sqrt {\frac{0.256(1-0.256)}{199}}=0.205$
$Upper~bound=p ̂+z_{\frac{α}{2}}.\sqrt {\frac{p ̂(1-p ̂)}{n}}=0.256+1.645\times\sqrt {\frac{0.256(1-0.256)}{199}}=0.307$
