Question

An Associated Press article on potential violent behavior reported the results of a survey of 750 workers who were employed full time (San Luis Obispo Tribune, September 7 , 1999). Of those surveyed, 125 indicated that they were so angered by a coworker during the past year that they felt like hitting the coworker (but didn't). Assuming that it is reasonable to regard this sample as representative of the population of full-time workers, use this information to construct and interpret a \(90 \%\) confidence interval estimate of \(p,\) the proportion of all full-time workers so angered in the last year that they wanted to hit a coworker.

Short answer

The \(90 \%\) confidence interval estimate of the proportion of full-time workers who wanted to hit a coworker out of anger in the last year is \([0.1407, 0.1927]\), or equivalently, between 14.07% and 19.27%.

Step-by-step solution

Calculate the Sample Proportion (\(p\))

The sample proportion (\(p\)) is calculated as the number of successful outcomes (workers who wanted to hit a coworker) divided by the total number of outcomes (total workers surveyed). So, \(p = 125/750 = 0.1667\)

Calculate the Standard Error (SE)

The Standard Error (SE) for a proportion can be calculated using the formula \(\sqrt{p(1-p)/n}\), where \(n\) is the sample size. Substituting \(p = 0.1667\) and \(n = 750\), we find \(SE = \sqrt{0.1667 * (1 - 0.1667) / 750} = 0.0137\).

Determine the Z-Score for a 90% Confidence Level

A 90% confidence level corresponds to a z-score of approximately 1.645. This value can be found in standard z-tables which provides the z-score associated with the desired level of confidence.

Calculate the Confidence Interval

The confidence interval can be calculated using the formula \(p ± (z*SE)\), where \(z\) is the z-score and SE is the standard error. So, the confidence interval becomes \(0.1667 ± (1.645*0.0137) = [0.1407, 0.1927]\). This means that we can be 90% confident that the true proportion of all full-time workers who got so angered in the past year that they wanted to hit a coworker lies between 14.07% and 19.27%.