Employees with substance abuse problems. According to the New Jersey Governor’s Council for a Drug-Free Workplace Report, 50 of the 72 sampled businesses that are members of the council admitted that they had employees with substance abuse problems. At the time of the survey, 251 New Jersey businesses were members of the Governor’s Council. Use the finite population correction factor to find a 95% confidence interval for the proportion of all New Jersey Governor’s Council business members who have employees with substance abuse problems. Interpret the resulting interval.

Let N denotes the business were member of the Governor’s Council, i.e. $N=251$

Let x denotes the number of the council admitted that they had employees, i.e. $x=50$

The sample size is $n=72$

Since, the proportion of all business were member of the Governor’s Council business members who have employee with substance abuse problems is,

$\hat{p}=\frac{x}{n}$

Therefore,

$\begin{array}{rcl}\hat{p}& =& \frac{50}{72}\\ & =& 0.6944\\ & & \end{array}$

The 95% confidence intervals obtained by using the following formula:

$\begin{array}{rcl}\hat{p}\pm 2\left(\sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}\times \sqrt{\frac{N-n}{N}}\right)& =& 0.6944\pm 2\left(\sqrt{\frac{0.6944\left(1-0.6944\right)}{72}}\times \sqrt{\frac{251-72}{251}}\right)\\ & =& 0.6944\pm 2\left(\sqrt{\frac{0.6944\times 0.3056}{72}}\times \sqrt{\frac{179}{251}}\right)\\ & =& 0.6944\pm 2\left(\sqrt{0.0029473}\times \sqrt{0.713147}\right)\\ & =& 0.6944\pm 0.0916\\ & =& \left(0.6028,0.7860\right)\\ & & \end{array}$

Hence, it is 95% confident that, the true proportion of all New Jersey Governor Council business members who have employee with substance abuse problem lies in between 0.6028 and 0.7860.