Question

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

The 95% confidence interval for the proportion of the entire population that could correctly describe the Bill of Rights is calculated to be \(p \pm \)\(Z*\)\(SE\). Fill in the calculated \(p\) and calculated \(SE\) to get the interval.

Step-by-step solution

Calculate the Sample Proportion

The sample proportion \(p\) is calculated as the number of successes (correct responses) divided by the sample size (total responses). Here, the successes are 142 (correct descriptions of the Bill of Rights), and the sample size is 507 people. So, \(p = \frac{142}{507}\).

Calculate the Standard Error

The standard error for the sample proportion \(p\) is calculated using the formula \(\sqrt{ \frac{p(1-p)}{n}} \), where \(n\) is the sample size. In this case, \(n = 507\) and \(p\) is the earlier calculated sample proportion.

Calculate the Confidence Interval

The formula for the confidence interval is \(p \pm \)\(Z*\)\(SE\), where \(Z*\) is the Z-value from a standard Normal distribution for the desired confidence level (for a 95% confidence interval, \(Z* = 1.96\)), and \(SE\) is the standard error calculated in Step 2. Plug the calculated values into this formula to obtain the confidence interval.