Question

Is the environment a major issue with Americans? To answer that question, a researcher conducts a survey of 1255 randomly selected Americans. Suppose 734 of the sampled people replied that the environment is a major issue with them. Construct a 95% confidence interval to estimate the proportion of Americans who feel that the environment is a major issue with them. What is the point estimate of this proportion?

Appendix A Statistical Tables

(Round the intermediate values to 3 decimal places. Round your answer to 3 decimal places.)

enter the lower limit of the confidence interval ≤ p ≤ enter the upper limit of the confidence interval  

The point estimate is enter the point estimate  .

Answer 1

$$ Sample size $ (n) = 1255 \\ $ Number of successes $ (x) = 734 \\\\ $ Point estimate $ = $ Sample proportion $ (\hat{p}) = \frac{x}{n} = \frac{734}{1255} \approx \mathbf{0.585} \\\\ \\ $ Standard error $ (SE) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.585(1-0.585)}{1255}} \approx 0.014 \\\\\\ $ Confidence Level $ = 95\% = 0.95 \\ \Rightarrow $ Significance Level $ (\alpha) = 1 - 0.95 = 0.05 \\ $ Critical value $ : z_{c} = z_{\alpha/2} = z_{0.05/2} =z _{0.025} = 1.96 \\ ( P(z < -1.96) = 0.025 $ from Z table $) \\\\ $ Margin of error $ (ME) = z_{c} * SE = 1.96 * 0.014 \approx 0.027 \\\\ $ Lower Limt $ = \hat{p} - ME = 0.585 - 0.027 = 0.558 \\\\ $ Upper Limit $ = \hat{p} &plus; ME = 0.585 &plus; 0.027 = 0.612 \\\\ $ So, 95\% confidence interval is $ \mathbf{0.558 < p < 0.612}$