A sample of 37 Charleston County households have a mean income of $ 32,449$ with a standard deviation of $ 2,954

Find a $95%confidence interval for the true population mean income for households in Charleston County.
Round your answers to the nearest dollar.

Step 1 ：We are given a sample of 37 Charleston County households with a mean income of $32,449 and a standard deviation of $2,954. We are asked to find a 95% confidence interval for the true population mean income for households in Charleston County.

Step 2 ：To solve this problem, we need to use the formula for the confidence interval for the population mean. The formula is given by: \[\bar{x} \pm z \frac{s}{\sqrt{n}}\] where: \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the desired confidence level, \(s\) is the standard deviation of the sample, and \(n\) is the sample size.

Step 3 ：In this case, we have \(\bar{x} = 32449\), \(s = 2954\), and \(n = 37\). The z-score for a 95% confidence interval is approximately 1.96.

Step 4 ：We can plug these values into the formula to find the confidence interval. The margin of error is calculated as \(z \frac{s}{\sqrt{n}}\), which gives us approximately 951.84.

Step 5 ：We then subtract this margin of error from the mean to get the lower bound of the confidence interval, and add it to the mean to get the upper bound. This gives us a lower bound of $31,497 and an upper bound of $33,401.

Step 6 ：Final Answer: The 95% confidence interval for the true population mean income for households in Charleston County is \(\boxed{(\$31,497, \$33,401)}\).