Step 1 :First, we are given the following values: the sample size \(n = 193\), the sample mean \(\overline{x} = \$45267\), the standard deviation \(\sigma = \$955\), and the confidence level \(0.96\).
Step 2 :We calculate the Z-score for a 96% confidence level. The Z-score is approximately \(2.0537489106318225\).
Step 3 :Next, we calculate the margin of error using the formula \(E = Z \times \frac{\sigma}{\sqrt{n}}\). The margin of error is approximately \$141.18.
Step 4 :Finally, we calculate the confidence interval using the formula \((\overline{x} - E, \overline{x} + E)\). The confidence interval is approximately (\$45125.82, \$45408.18).
Step 5 :This means that we are 96% confident that the true average household income in St. Lucie county is between \$45125.82 and \$45408.18.
Step 6 :Final Answer: The margin of error is \(\boxed{\$141.18}\). A 96% confidence interval for the average household income in St. Lucie county is \(\boxed{(\$45125.82, \$45408.18)}\).