In a random sample of 100 audited estate tax returns, it was determined that the mean amount of additional tax owed was $\$ 3425$ with a standard deviation of $\$ 2586$. Construct and interpret a $90 \%$ confidence interval for the mean additional amount of tax owed for estate tax returns. Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2). Click here to view the table of critical t-values. Find and interpret a $90 \%$ confidence interval for the mean additional amount of tax owed for estate tax returns. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to the nearest dollar as needed.) A. There is a $90 \%$ probability that the mean additional tax owed is between $\$ \square$ and $\$ \square$. B. One can be $90 \%$ confident that the mean additional tax owed is between $\$ \square$ and $\$ \square$. C. $90 \%$ of taxes owed for estate tax returns are between $\$ \square$ and $\$ \square$.
Solution
Step 1 ：Given in the problem, we have: \(\bar{x} = $3425\), \(\sigma = $2586\), and \(n = 100\).
Step 2 ：The Z-score for a 90% confidence interval is 1.645 (you can find this value in a standard normal distribution table).
Step 3 ：Substitute these values into the formula for a confidence interval: \(CI = \bar{x} ± Z * (\sigma/\sqrt{n})\)
Step 4 ：Calculate the standard deviation divided by the square root of the sample size: \(\sigma/\sqrt{n} = $2586/\sqrt{100} = $258.6\)
Step 5 ：Multiply the Z-score by the result from step 4: \(1.645 * $258.6 = $425.63\)
Step 6 ：Add and subtract the result from step 5 to the sample mean to get the confidence interval: \(CI = $3425 ± $425.63\)
Step 7 ：\(\boxed{CI = [$3000, $3851]}\). So, one can be 90% confident that the mean additional tax owed is between $3000 and $3851.
