Questions: In a survey of 2272 adults, 750 say they believe in UFOs. Construct a 99% confidence interval for the population proportion of adults who believe in UFOs. A 99% confidence interval for the population proportion is (0.305, 0.356). (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 99% of adults believe in UFOs. B. With 99% probability, the population proportion of adults who do not believe in UFOs is between the endpoints of the given confidence interval. C. With 99% confidence, it can be said that the population proportion of adults who believe in UFOs is between the endpoints of the given confidence interval. D. With 99% confidence, it can be said that the sample proportion of adults who 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:

\[ \hat{p} = \frac{x}{n} = \frac{750}{2272} \approx 0.330 \]

To construct a 99% 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 99%, \( z \approx 2.576 \)),
• \( n \) is the sample size.

Substituting the values:

\[ 0.330 \pm 2.576 \cdot \sqrt{\frac{0.330(1 - 0.330)}{2272}} \]

Calculating the margin of error:

\[ \sqrt{\frac{0.330(1 - 0.330)}{2272}} \approx 0.0257 \]

Thus, the confidence interval becomes:

\[ 0.330 \pm 2.576 \cdot 0.0257 \approx 0.330 \pm 0.066 \]

This results in:

\[ (0.305, 0.356) \]

With 99% confidence, it can be stated that the population proportion of adults who believe in UFOs is between the endpoints of the given confidence interval:

\[ \text{Confidence Interval: } (0.305, 0.356) \]

Final Answer