Questions: Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. n=195, x=162; 95% confidence A. 0.777<p<0.884 B. 0.778<p<0.883 C. 0.789<p<0.873 D. 0.788<p<0.873

Solution Steps

The sample proportion \( \hat{p} \) is calculated as: \[ \hat{p} = \frac{x}{n} = \frac{162}{195} \approx 0.8308 \]

For a 95% confidence interval, the critical value \( z_{\alpha/2} \) is approximately 1.96 (from the standard normal distribution table).

The margin of error \( E \) is calculated using the formula: \[ E = z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] Substitute the values: \[ E = 1.96 \cdot \sqrt{\frac{0.8308(1-0.8308)}{195}} \approx 1.96 \cdot \sqrt{\frac{0.8308 \cdot 0.1692}{195}} \approx 1.96 \cdot \sqrt{0.000719} \approx 1.96 \cdot 0.0268 \approx 0.0525 \]

The confidence interval for the population proportion \( p \) is: \[ \hat{p} - E < p < \hat{p} + E \] Substitute the values: \[ 0.8308 - 0.0525 < p < 0.8308 + 0.0525 \] \[ 0.7783 < p < 0.8833 \]

The calculated confidence interval \( 0.7783 < p < 0.8833 \) matches option B: \( 0.778 < p < 0.883 \).

Final Answer