Question

A management professor was in the process of investigating the relationship between education and managerial level achieved. The source of his data was a survey of 385 CEOs of medium and large companies. He discovered that there was only one CEO who did not have at least one university degree. Estimate (using a Wilson estimator) with 99% confidence the proportion of CEOs of medium and large companies with no university degrees.

Answer 1

A management professor was in the process of investigating the relation between education and managerial level achieved. The source of his data was a survey of 385 CEOs of medium and large companies. He discovered only one CEO who did not have at least one university degree.

So from the given information we have \(n=385, x=1\)

The Wilson estimate is

\(\begin{aligned} \tilde{p} &=\frac{x+2}{n+4} \\ &=\frac{1+2}{385+4} \\ &=0.0077 \end{aligned}\)

Now the \(99 \%\) confidence interval estimate of the proportion of spam e-mails that get through is

$$ \tilde{p} \pm z_{\alpha / 2} \sqrt{\frac{\tilde{p}(1-\tilde{p})}{n+4}} $$

By substituting the value we get

\(0.0077 \pm 2.58 \sqrt{\frac{0.0077(1-0.0077)}{385+4}}\)

\((-0.0037,0.0191)\)

Therefore the \(99 \%\) confidence interval estimate of the proportion of CEOs of medium and large companies with no university degrees is between \(-0.0037\) and \(0.0191\).