(Solved): To construct a confidence interval for the difference between two population means \( \mu_{1}-\mu_{ ...

To construct a confidence interval for the difference between two population means \( \mu_{1}-\mu_{2} \), use the formula shown below when both population standard deviations are known, and either both populations are normally distributed or both \( n_{1} \geq 30 \) and \( n_{2} \geq 30 \). Also, the samples must be randomly selected and independent. \[ \left(\bar{x}_{1}-\bar{x}_{2}\right)-z_{c} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}<\mu_{1}-\mu_{2}<\left(\bar{x}_{1}-\bar{x}_{2}\right)+z_{c} \sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}} \] The descriptive statistics for the annual salaries from a random sample of microbiologists from two regions are shown below. Construct a \( 95 \% \) confidence interval for the difference between the mean annual salaries. \[ \bar{x}_{1}=\$ 105,430, n_{1}=45, \text { and } \sigma_{1}=\$ 8930 ; \bar{x}_{2}=\$ 85,190, n_{2}=38, \text { and } \sigma_{2}=\$ 8940 \] Complete the \( 95 \% \) confidence interval for \( \mu_{1}-\mu_{2} \) below. \( \$<\mu_{1}-\mu_{2}<\$ \) (Round to the nearest dollar as needed.)