Let's solve this problem step by step.

Problem Summary:

We are given:

• Sample size ($n$) = 100
• Mean additional tax owed ($\bar{x}$) = $3,409
• Standard deviation ($s$) = $2,559
• Confidence level = 90%

We need to construct and interpret a 90% confidence interval for the mean additional tax owed.

Formula for Confidence Interval:

$\text{Confidence Interval} = \bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ Where:

• $\bar{x}$ is the sample mean,
• $z_{\alpha/2}$ is the critical value for the confidence level (for 90%, $z_{\alpha/2} \approx 1.645$),
• $s$ is the sample standard deviation,
• $n$ is the sample size.

Step 1: Calculate the Margin of Error

$\text{Margin of Error} = z_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 1.645 \cdot \frac{2559}{\sqrt{100}} = 1.645 \cdot 255.9 \approx 420.46$

Step 2: Calculate the Confidence Interval

$\text{Lower Bound} = \bar{x} - \text{Margin of Error} = 3409 - 420.46 \approx 2989$ $\text{Upper Bound} = \bar{x} + \text{Margin of Error} = 3409 + 420.46 \approx 3830$

Final Confidence Interval:

$\boxed{(2989, 3830)}$

Correct Interpretation (Option A):

One can be 90% confident that the mean additional tax owed is between $2989 and $3830.

Let me know if you'd like further details or have any questions!

Here are 5 related questions:

1. How would the confidence interval change if the sample size increased?
2. What happens to the confidence interval if the standard deviation decreases?
3. How do different confidence levels (e.g., 95%) affect the confidence interval width?
4. Why is the Z-value used in this case rather than the t-value?
5. What are the assumptions for constructing a confidence interval using this method?

Tip: Always double-check critical values based on the confidence level when using Z-distribution tables!