Question

A Rasmussen Reports survey of 1,000 US adults found that 42% believe raising the minimum wage will help the economy. Construct a 99% confidence interval for the true proportion of US adults who believe this.

Answer 1

Answer:

(0.3798, 0.4602)

Step-by-step explanation:

Let p be the true proportion of US adults who believe raising the minimum wage will help the economy. A point estimate for p is $\hat{p}=0.42$ and a good aproximation (because we have a large sample) for the standard deviation of $\hat{p}$ is $\sqrt{\hat{p}(1-\hat{p})/n}=\sqrt{0.42(0.58)/1000}=0.0156$. Therefore, a 99% confidence interval for p is given by $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ where $\alpha=0.01$ and $z_{\alpha/2}$ is the $\alpha/2$th quantile of the standard normal distribution, so we have, $0.42\pm z_{0.005}0.0156$, i.e., $0.42\pm(-2.5758)(0.0156)$ or equivalently (0.3798, 0.4602).