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 of 750 as a random sample from the population of full-time workers, use this information to construct and interpret a \(90 \%\) confidence interval estimate of \(\pi\), the true proportion of fulltime workers so angered in the last year that they wanted to hit a colleague.

Short answer

The 90% confidence interval for the true proportion of full-time workers so angered in the past year that they wanted to hit a colleague is approximately from 14.44% to 18.90%.

Step-by-step solution

Calculate the Sample Proportion

The sample proportion (\( p̂ \)) is the proportion of individuals in the random sample who have been so angered. This can be calculated by taking the number of angered individuals and dividing by the total number of people surveyed. \( p̂ =\frac{number\ of\ angered\ respondents}{total\ number\ of\ respondents} =\frac{125}{750}=0.1667 \)

Determine the z-Score

A z-score reflects how many standard deviations away our statistic is from the mean. For a 90% confidence interval, we look up the z-score that leaves 90% in the middle and 10% in the two tails of a standard normal distribution. This z-score is approximately 1.645.

Calculate the Standard Error

The standard error (SE) is a measure of how spread out numbers are in statistical data. It can be computed using the formula:\( SE=\sqrt{ (p̂ * (1 - p̂))/ n } =\sqrt{(0.1667 * (1 - 0.1667))/ 750} = 0.0135 \)

Construct the Confidence Interval

The 90% confidence interval can now be calculated:\( CI = p̂ ± (z * SE) \)Lower limit = \(0.1667 - (1.645 * 0.0135) = 0.1444\)Upper limit = \(0.1667 + (1.645 * 0.0135) = 0.1890\)

Interpret the Confidence Interval

We can be 90% confident that the true proportion of full-time workers who felt like hitting a coworker out of anger in the past year lies between 14.44% and 18.90%.