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Chapter 9: Problem 70

The Associated Press (December 16,1991 ) reported that in a random sample of 507 people, only 142 correctly described the Bill of Rights as the first 10 amendments to the U.S. Constitution. Calculate a \(95 \%\) confidence interval for the proportion of the entire population that could give a correct description.

Short Answer

Expert verified
The 95% confidence interval for the proportion of the entire population that could correctly describe the Bill of Rights is estimated to be between 0.23642 and 0.32374.

Step by step solution

01

Calculate the Sample Proportion

The sample proportion \(p\) is the ratio of successful trials to the total number of trials. In this case, it will be the number of people who correctly described the Bill of Rights divided by the total sample size. So, \(p = 142 / 507 = 0.28008\).
02

Standard Error of the Sample Proportion

Standard error, an indication of how much our sample proportion is likely to vary from our actual population proportion, can be calculated by the formula \(SE = \sqrt{p(1-p)/n}\), where \(p\) is the sample proportion we computed in Step 1 and \(n\) is the total size of our sample. So, \(SE = \sqrt{0.28008 * (1 - 0.28008) / 507} = 0.02224\).
03

Calculate the 95% Confidence Interval

For a 95% confidence interval, the Z-value (which comes from the standard normal distribution) is approximately 1.96. The formula to calculate the interval is \(p \pm Z*SE\). So the interval will be (0.28008 - 1.96 * 0.02224, 0.28008 + 1.96 * 0.02224) = (0.23642, 0.32374).

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Most popular questions from this chapter

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