In a survey of 2328 adults, 725 say they believe in UFOs.
Construct a $90 \%$ confidence interval for the population proportion of adults who believe in UFOs.
A $90 \%$ confidence interval for the population proportion is (Round to three decimal places as needed.)
Answer
\(\boxed{\text{Final Answer: The 90% confidence interval for the population proportion of adults who believe in UFOs is approximately (0.296, 0.327).}}\)
Step 1 ：First, calculate the sample proportion by dividing the number of successes (the number of adults who believe in UFOs) by the total number of observations (the total number of adults surveyed). In this case, the number of successes is 725 and the total number of observations is 2328. So, the sample proportion \(p\) is \(\frac{725}{2328} = 0.311\).
Step 2 ：Next, calculate the standard error of the proportion. The formula for the standard error of a proportion is \(\sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion and \(n\) is the total number of observations. Substituting the values we have, the standard error is \(\sqrt{\frac{0.311(1-0.311)}{2328}} = 0.010\).
Step 3 ：Then, calculate the margin of error. The Z-score for a 90% confidence interval is 1.645. The margin of error is the Z-score times the standard error, which is \(1.645 \times 0.010 = 0.016\).
Step 4 ：Finally, construct the confidence interval. The confidence interval is the sample proportion plus and minus the margin of error. So, the 90% confidence interval for the population proportion of adults who believe in UFOs is \((0.311 - 0.016, 0.311 + 0.016)\).
Step 5 ：\(\boxed{\text{Final Answer: The 90% confidence interval for the population proportion of adults who believe in UFOs is approximately (0.296, 0.327).}}\)
