Questions: In a survey of 2303 adults, 715 say they believe in UFOs. Construct a 95% confidence interval for the population proportion of adults who believe in UFOs. A 95% confidence interval for the population proportion is ( (Round to three decimal places as needed.) Interpret your results. Choose the correct answer below. A. The endpoints of the given confidence interval shows that 95% of adults believe in UFOs. B. With 95% confidence, it can be said that the population proportion of adults who believe in UFOs is between the endpoints of the given confidence interval. C. With 95% confidence, it can be said that the sample proportion of adults who believe in UFOs is between the endpoints of the given confidence interval. D. With 95% probability, the population proportion of adults who do not believe in UFOs is between the endpoints of the given confidence interval.

Solution Steps

The sample proportion of adults who believe in UFOs is calculated as follows:

\[ \hat{p} = \frac{x}{n} = \frac{715}{2303} \approx 0.3105 \]

To construct a 95% confidence interval for the population proportion, we use the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where:

• \( \hat{p} \) is the sample proportion.
• \( z \) is the z-score corresponding to the desired confidence level (for 95%, \( z \approx 1.96 \)).
• \( n \) is the sample size.

Substituting the values:

\[ 0.3105 \pm 1.96 \cdot \sqrt{\frac{0.3105(1 - 0.3105)}{2303}} \]

Calculating the margin of error:

\[ \text{Margin of Error} = 1.96 \cdot \sqrt{\frac{0.3105 \cdot 0.6895}{2303}} \approx 0.0185 \]

Thus, the confidence interval is:

\[ (0.3105 - 0.0185, 0.3105 + 0.0185) = (0.292, 0.329) \]

With 95% confidence, it can be said that the population proportion of adults who believe in UFOs is between the endpoints of the given confidence interval:

\[ \text{Confidence Interval: } (0.292, 0.329) \]

Final Answer