Math Problem Statement

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.)

Solution

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 (p^\hat{p})
  2. The sample size (nn)

The formula for the confidence interval for a proportion is:

p^±Zα/2p^(1p^)n\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

where:

  • p^\hat{p} is the sample proportion
  • Zα/2Z_{\alpha/2} is the critical value from the standard normal distribution for a 95% confidence level, which is 1.96
  • nn is the sample size

Let's assume we have the sample proportion and sample size:

  • p^=0.55\hat{p} = 0.55 (example value)
  • n=1000n = 1000 (example value)

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

SE=p^(1p^)n=0.55(10.55)1000=0.550.451000=0.24751000=0.00024750.0157SE = \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=Zα/2SE=1.960.01570.0308ME = Z_{\alpha/2} \cdot SE = 1.96 \cdot 0.0157 \approx 0.0308

Finally, we can calculate the confidence interval:

p^±ME=0.55±0.0308\hat{p} \pm ME = 0.55 \pm 0.0308

So, the 95% confidence interval is:

(0.519,0.581)(0.519, 0.581)

Please adjust the values of p^\hat{p} and nn 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 (p^\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.

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Math Problem Analysis

Mathematical Concepts

Statistics
Confidence Intervals
Sampling

Formulas

Confidence interval formula for proportions

Theorems

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Suitable Grade Level

Advanced High School