Step 1 :Given values are sample means \(\bar{x}_{1}=5318\) and \(\bar{x}_{2}=5284\), standard deviations \(s_{1}=142\) and \(s_{2}=190\), and sample sizes \(n_{1}=n_{2}=395\). The z-score for a 95% confidence level is 1.96.
Step 2 :Calculate the standard error (SE) using the formula \(SE = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}\). Substituting the given values, we get \(SE = \sqrt{\frac{142^{2}}{395} + \frac{190^{2}}{395}} = 11.934844210508738\).
Step 3 :Calculate the confidence interval (CI) using the formula \(CI = (\bar{x}_{1} - \bar{x}_{2}) \pm z*SE\). Substituting the given values, we get \(CI = (5318 - 5284) \pm 1.96*11.934844210508738\).
Step 4 :Calculate the lower and upper bounds of the confidence interval. The lower bound is \(CI_{lower} = (5318 - 5284) - 1.96*11.934844210508738 = 10.607705347402874\) and the upper bound is \(CI_{upper} = (5318 - 5284) + 1.96*11.934844210508738 = 57.39229465259713\).
Step 5 :Final Answer: The 95% confidence interval for the difference between the population means \(\mu_{1} - \mu_{2}\) is \(\boxed{(10.6, 57.4)}\). This means we are 95% confident that the true difference between the population means falls within this interval.