Question

use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.

n = 56, x = 30, 85% confidence

o a. 0.425 < p < 0.647
o b. 0.405 < p < 0.666
o c. 0.404 < p < 0.669
o d. 0.428 < p < 0.646

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$. Given $x = 30$ and $n=56$, so $\hat{p}=\frac{30}{56}\approx0.536$.

Step2: Find critical value

For a $95\%$ confidence interval, the significance level $\alpha=1 - 0.95=0.05$, and $\alpha/2=0.025$. The critical value $z_{\alpha/2}=z_{0.025} = 1.96$.

Step3: Calculate margin of error

The margin of error $E=z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.536$, $n = 56$ and $z_{\alpha/2}=1.96$ into the formula:

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Step4: Construct confidence interval

The confidence interval for the population proportion $p$ is $\hat{p}-E$0.536-0.131

Answer:

B. \(0.405 < p<0.666\)