Question

in a random sample of 100 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3450 with a standard deviation of $2520. 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 $□ and $□
○ b. 90% of taxes owed for estate tax returns are between $□ and $□
○ c. one can be 90% confident that the mean additional tax owed is between $□ and $□

Explanation:

Step1: Determine the critical value

For a 90% confidence interval, the significance level \(\alpha=1 - 0.90=0.10\), and \(\alpha/2 = 0.05\). Since \(n = 100\) (a large sample, \(n\geq30\)), we use the standard normal distribution \(z\). 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 \(s = 2520\) and \(n = 100\).

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Step3: Calculate the confidence interval

The formula for the confidence interval for the population mean \(\mu\) (when \(n\) is large) is \(\bar{x}-E<\mu<\bar{x} + E\), where \(\bar{x}=3450\).

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Rounding to the nearest dollar, the confidence interval is \(3035<\mu<3865\).

The interpretation of a confidence interval: A confidence interval gives a range of values within which we are confident the population parameter lies. For a 90% confidence interval for the mean, we can say that one can be 90% confident that the mean additional tax owed is within the calculated interval.

Answer:

C. One can be 90% confident that the mean additional tax owed is between \(\boldsymbol{3035}\) and \(\boldsymbol{3865}\)