Let's break down each part of the problem step-by-step:

Part 1: Number of people who felt this way

Given:

• Percentage of adults who believe in protecting the rights of those with unpopular views: $81\%$
• Sample size: $900$

To find the number of people: $\text{Number of people} = \hat{p} \times n = 0.81 \times 900 = 729$

Part 2: Central Limit Theorem Application

To apply the Central Limit Theorem, we need to check the conditions. The conditions require both the expected number of successes ($np$) and the expected number of failures ($n(1-p)$) to be at least 10.

Given:

• Sample size ($n$): 900
• Sample proportion ($\hat{p}$): 0.81

Expected number of successes: $np = 900 \times 0.81 = 729$

Expected number of failures: $n(1 - p) = 900 \times (1 - 0.81) = 900 \times 0.19 = 171$

Both 729 and 171 are greater than 10, so the sample size is large enough to apply the Central Limit Theorem.

Therefore, the correct choice is: B. Yes, the sample size is large enough, since the expected number of successes is 729 and the expected number of failures is 171, both of which are greater than or equal to 10.

Part 3: 95% Confidence Interval

Given:

• Sample proportion ($\hat{p}$): 0.81
• Sample size ($n$): 900

First, calculate the standard error (SE): $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.81 \times 0.19}{900}} = \sqrt{\frac{0.1539}{900}} = \sqrt{0.000171} \approx 0.0131$

Next, calculate the margin of error (ME) using the critical value for a 95% confidence level ($Z_{\alpha/2}$ = 1.96): $ME = Z_{\alpha/2} \times SE = 1.96 \times 0.0131 \approx 0.0257$

Finally, calculate the confidence interval: $\hat{p} \pm ME = 0.81 \pm 0.0257$ $\text{Lower limit} = 0.81 - 0.0257 = 0.7843$ $\text{Upper limit} = 0.81 + 0.0257 = 0.8357$

So, the 95% confidence interval is: $(0.784, 0.836)$

Would you like further details or have any questions?

Here are five questions you might ask next:

1. How do I interpret the confidence interval in context?
2. What if the sample size was different? How would that change the confidence interval?
3. How can I calculate a confidence interval for a different confidence level, like 99%?
4. What is the relationship between the confidence interval and the margin of error?
5. How can I determine if my sample proportion is significantly different from a hypothesized population proportion?

Tip: When interpreting a confidence interval, remember that it provides a range of values within which the true population parameter is likely to fall, given a certain level of confidence.