In a survey of 3998 adults, 725 oppose allowing transgender students to use the bathrooms of the opposite biological sex.
Construct a $99 \%$ confidence interval for the population proportion. Interpret the results.
A $99 \%$ confidence interval for the population proportion is (Round to three decimal places as needed.)
Answer
\(\boxed{\text{Final Answer: The 99% confidence interval for the population proportion is (0.166, 0.197). This means that we are 99% confident that the true population proportion of adults who oppose allowing transgender students to use the bathrooms of the opposite biological sex is between 16.6% and 19.7%.}}\)
Step 1 ：First, we calculate the sample proportion by dividing the number of adults who oppose allowing transgender students to use the bathrooms of the opposite biological sex (725) by the total number of adults surveyed (3998). This gives us a sample proportion of approximately 0.181.
Step 2 ：Next, we calculate the standard error of the proportion using the formula \(\sqrt{p(1 - p) / n}\), where p is the sample proportion and n is the total number of observations. Substituting our values in, we get a standard error of approximately 0.006.
Step 3 ：We then use the Z-score for a 99% confidence interval, which is approximately 2.576, to calculate the margin of error. The margin of error is the Z-score times the standard error, giving us a margin of error of approximately 0.016.
Step 4 ：Finally, we construct the confidence interval by adding and subtracting the margin of error from the sample proportion. This gives us a 99% confidence interval for the population proportion of approximately (0.166, 0.197).
Step 5 ：\(\boxed{\text{Final Answer: The 99% confidence interval for the population proportion is (0.166, 0.197). This means that we are 99% confident that the true population proportion of adults who oppose allowing transgender students to use the bathrooms of the opposite biological sex is between 16.6% and 19.7%.}}\)
