Question

a stats class was interested in determining the percentage of students on their campus who are opposed to a new policy eliminating all paper books from the campus bookstore in favor of ebooks. after surveying 186 students, the class finds that 33% are opposed to the policy. the margin of error is 6.9%. if surveying more students keeps the percentage at 33%, how many more would need to be surveyed in order to be 95% confident that the correct percentage for the entire student body is between 30% and 36%? dont forget to consider the number of students that have already been surveyed. round up to the nearest student, if necessary. (hint: what would the margin of error be in this case?) in order to be 95% confident that the correct percentage for the entire student body is between 30% and 36%, the class needs to survey more students.

Explanation:

Step1: Find the new margin of error

The desired interval is \(30\%\) to \(36\%\) with a sample proportion \(p = 33\%\). The margin of error \(E\) is \(\frac{36 - 30}{2}=3\%=0.03\)

Step2: Use the formula for margin of error

The formula for margin of error for a proportion is \(E = z\sqrt{\frac{p(1 - p)}{n}}\). For a \(95\%\) confidence interval, \(z = 1.96\), \(p=0.33\), and \(1 - p = 0.67\)

We want to solve for \(n\) in the formula \(E=z\sqrt{\frac{p(1 - p)}{n}}\). Rearranging the formula for \(n\) gives \(n=\frac{z^{2}p(1 - p)}{E^{2}}\)

Substitute \(z = 1.96\), \(p = 0.33\), \(1 - p=0.67\), and \(E = 0.03\) into the formula:

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Step3: Calculate the number of additional students

Since \(186\) students have already been surveyed, the number of additional students \(N\) is \(n-186\)

\(N=944 - 186=758\)

Answer:

\(758\)