Answer
a)0.82
b)p is between 0.82-0.0578=0.7622 and 0.82+0.0578=0.8778.
c)The method is effective.
Work Step by Step
a) The best point estimate is equal to the proportion of the sample (x) divided by the sample size: $\hat{p}=\frac{x}{n}=\frac{239}{291}=0.82.$
b)$E=z_{\frac{\alpha}{2}}\cdot \sqrt{\frac{\hat{p}\cdot (1-\hat{p})}{n}}=2.575\cdot \sqrt{\frac{0.82\cdot (1-0.82)}{291}}=0.0578.$
Hence, the confidence interval: E is between $\hat{p}-E$ and $\hat{p}+E$, hence p is between 0.82-0.0578=0.7622 and 0.82+0.0578=0.8778.
c) The method is effective, because without the method, the proportion would be approximately 50%, whilst here, according to b), 50% is much lower than the lower bound of the confidence interval.
