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To calculate the 90% confidence interval for the mean additional tax owed, we use the formula for the confidence interval of the mean:

\[
CI = \bar{x} \pm z_{\alpha/2} \cdot \left( \frac{\sigma}{\sqrt{n}} \right)
\]

Where:
- $\bar{x} = 3413$ (the sample mean)
- $\sigma = 2535$ (the standard deviation)
- $n = 81$ (the sample size)
- $z_{\alpha/2}$ is the critical value from the standard normal distribution for a 90% confidence level, which is approximately 1.645.

Now, calculating the margin of error (MOE):

\[
MOE = 1.645 \cdot \left( \frac{2535}{\sqrt{81}} \right) = 1.645 \cdot \left( \frac{2535}{9} \right) = 1.645 \cdot 281.67 \approx 463.55
\]

Thus, the confidence interval is:

\[
CI = 3413 \pm 463.55
\]

This gives:

\[
\text{Lower limit} = 3413 - 463.55 = 2949.45 \approx 2950
\]
\[
\text{Upper limit} = 3413 + 463.55 = 3876.55 \approx 3876
\]

Therefore, the 90% confidence interval is approximately between **$2950** and **$3876**.

The correct choice in the multiple-choice options is:

- C: "One can be 90% confident that the mean additional tax owed is between $2950 and $3876."

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### Related questions:
1. What is the significance of the 90% confidence level in this context?
2. How would the confidence interval change with a larger sample size?
3. How does the standard deviation affect the width of the confidence interval?
4. What is the interpretation of a 95% confidence interval in comparison to a 90% interval?
5. Can we interpret a confidence interval as having a 90% probability that the true mean lies within the interval?

**Tip:** Confidence intervals provide a range of plausible values for the population mean, not a definitive range where the mean is guaranteed to fall.

In a random sample of 81 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3413 with a standard deviation of $2535. Construct and interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns.

Learn how to calculate the 90% confidence interval for the mean additional amount of tax owed based on a sample size of 81, a mean of $3413, and a standard deviation of $2535. Detailed step-by-step calculations are provided.

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