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) $0.778

Question

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) $0.778 < p < 0.883$
B) $0.788 < p < 0.873$
C) $0.789 < p < 0.873$
D) $0.777 < p < 0.884$

Quizplus · Accepted Answer

To construct a 95% confidence interval for the population proportion $p$, we use the formula: $\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $\hat{p} = \frac{x}{n}$ is the sample proportion, $n$ is the sample size, and $z$ is the z-score corresponding to the desired level of confidence. For a 95% confidence level, $z \approx 1.96$.Given $n = 195$ and $x = 162$, we find $\hat{p} = \frac{162}{195} = 0.8308$.The margin of error (ME) is calculated as: $ME = 1.96 \sqrt{\frac{0.8308(1-0.8308)}{195}} \approx 0.052$.Thus, the confidence interval is $0.8308 \pm 0.052$, which simplifies to $0.7788 < p < 0.8828$, or rounded, $0.778 < p < 0.883$.

Use the Given Degree of Confidence and Sample Data to Construct