Question

1. a survey of 866 voters in one state reveals that 47.1% of them favor approval of an issue before the legislature. construct a 95% confidence interval for the true proportion of all voters in the state who favor approval. express the c.i. as (____ ± ____). (0.438, 0.504), (.471 ±.033)

Explanation:

Step1: Find the critical value

For a 95% confidence interval, the critical value \(z_{\alpha/2}\) is \(1.96\).

Step2: Calculate the standard error

The sample proportion \(\hat{p}=0.471\), and \(n = 866\). The standard error \(SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.471\times(1 - 0.471)}{866}}\approx\sqrt{\frac{0.471\times0.529}{866}}\approx\sqrt{\frac{0.249159}{866}}\approx\sqrt{0.0002877}\approx0.01696\)

Step3: Calculate the margin of error

The margin of error \(E=z_{\alpha/2}\times SE = 1.96\times0.01696\approx0.0332\)

Answer:

\((0.471\pm0.033)\)