Question

in a random sample of 49 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3442 with a standard deviation of $2567. 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. 90% of taxes owed for estate tax returns are between $ and $
○ b. there is a 90% probability that the mean additional tax owed is between $ and $
○ c. one can be 90% confident that the mean additional tax owed is between $ and $

Explanation:

Step1: Determine the critical value

Since the sample size \(n = 49\) (\(n>30\)), we can use the standard normal distribution. For a \(90\%\) confidence interval, the significance level \(\alpha=1 - 0.90=0.10\), and \(\alpha/2=0.05\). From the standard - normal distribution table, \(z_{\alpha/2}=z_{0.05}\approx1.645\)

Step2: Calculate the standard error

The formula for the standard error \(SE=\frac{s}{\sqrt{n}}\), where \(s = 2567\) and \(n = 49\). So \(SE=\frac{2567}{\sqrt{49}}=\frac{2567}{7}=366.714\)

Step3: Calculate the margin of error

The margin of error \(E = z_{\alpha/2}\times SE\). Substitute \(z_{\alpha/2}=1.645\) and \(SE = 366.714\) into the formula. \(E=1.645\times366.714\approx603.25\)

Step4: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x} + E\), where \(\bar{x}=3442\).
\(\bar{x}-E=3442-603.25 = 2838.75\)
\(\bar{x}+E=3442 + 603.25=4045.25\)

Answer:

One can be \(90\%\) confident that the mean additional tax owed is between \(\$2839\) and \(\$4045\)