c. Find a​ 95% confidence interval for the proportion of adults in the country who believe that protecting the rights of those with unpopular views is a very important component of a strong democracy.
The​ 95% confidence interval is ​(

enter your response here​,

enter your response here​).
​(Round to three decimal places as​ needed.)

To find the 95% confidence interval for the proportion of adults in the country who believe that protecting the rights of those with unpopular views is a very important component of a strong democracy, we need the following information:

1. The sample proportion (\(\hat{p}\))
2. The sample size (\(n\))

The formula for the confidence interval for a proportion is:

\[
\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
\]

where:
- \(\hat{p}\) is the sample proportion
- \(Z_{\alpha/2}\) is the critical value from the standard normal distribution for a 95% confidence level, which is 1.96
- \(n\) is the sample size

Let's assume we have the sample proportion and sample size:
- \(\hat{p} = 0.55\) (example value)
- \(n = 1000\) (example value)

Now, we can calculate the standard error (SE):

\[
SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.55 \cdot (1-0.55)}{1000}} = \sqrt{\frac{0.55 \cdot 0.45}{1000}} = \sqrt{\frac{0.2475}{1000}} = \sqrt{0.0002475} \approx 0.0157
\]

Next, we find the margin of error (ME):

\[
ME = Z_{\alpha/2} \cdot SE = 1.96 \cdot 0.0157 \approx 0.0308
\]

Finally, we can calculate the confidence interval:

\[
\hat{p} \pm ME = 0.55 \pm 0.0308
\]

So, the 95% confidence interval is:

\[
(0.519, 0.581)
\]

Please adjust the values of \(\hat{p}\) and \(n\) to your specific problem's data.

Would you like further details or have any questions?

Here are five questions you might ask next:
1. How do I determine the sample proportion (\(\hat{p}\)) from my data?
2. What is the critical value for a 99% confidence interval?
3. How does sample size affect the width of the confidence interval?
4. Can you explain the concept of standard error in more detail?
5. How do I interpret this confidence interval in the context of my data?

**Tip:** The larger your sample size, the narrower your confidence interval will be, providing a more precise estimate of the population proportion.

Learn how to calculate a 95% confidence interval for the proportion of adults who believe protecting unpopular views is vital for democracy. This problem covers statistical concepts including confidence intervals and sampling methods. Ideal for advanced high school students.

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