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, \(\alpha=1 - 0.90=0.10\), and \(\alpha/2=0.05\). From the standard - normal distribution table, \(z_{\alpha/2}=z_{0.05}\approx1.645\)

Step2: Calculate the margin of error

The formula for the margin of error \(E\) is \(E = z_{\alpha/2}\frac{s}{\sqrt{n}}\), where \(s = 2567\), \(n = 49\), and \(z_{\alpha/2}=1.645\)

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Step3: Calculate the confidence interval

The formula for the confidence interval for the population mean \(\mu\) (when \(\sigma\) is unknown and \(n\geq30\)) is \(\bar{x}-E<\mu <\bar{x}+E\), where \(\bar{x}=3442\)

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Rounding to the nearest dollar, we get \(2839<\mu<4045\)

The interpretation of a confidence interval: A confidence interval gives a range of values within which the population parameter (in this case, the population mean \(\mu\) of the additional amount of tax owed) is likely to fall. A \(90\%\) confidence interval means that if we were to take many samples and construct confidence intervals in the same way, about \(90\%\) of those intervals would contain the true population mean.

Answer:

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