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To calculate the 90% confidence interval for the mean additional amount of tax owed, we'll need to follow these steps:

### Given Data:
- Sample mean ($\bar{x}$) = 3441
- Sample standard deviation ($s$) = 2533
- Sample size ($n$) = 49

### Step 1: Find the critical value (z-score)
Since we are using a 90% confidence level, we look up the z-score corresponding to a 90% confidence interval. This can be found in z-tables and is approximately 1.645.

### Step 2: Calculate the standard error (SE)
The standard error of the mean is calculated as:
\[
SE = \frac{s}{\sqrt{n}} = \frac{2533}{\sqrt{49}} = \frac{2533}{7} = 361.86
\]

### Step 3: Compute the margin of error (ME)
The margin of error is:
\[
ME = z \times SE = 1.645 \times 361.86 = 595.46
\]

### Step 4: Calculate the confidence interval
The confidence interval is given by:
\[
\text{CI} = \bar{x} \pm ME
\]
\[
\text{CI} = 3441 \pm 595.46
\]
Thus, the confidence interval is approximately:
\[
[3441 - 595, 3441 + 595] = [2846, 4036]
\]

### Final Answer:
Thus, the 90% confidence interval for the mean additional tax owed is between **2846** and **4036**.

Would you like more details or have any questions?

### Follow-up Questions:
1. How does changing the confidence level affect the confidence interval?
2. How is the sample size affecting the confidence interval?
3. What is the importance of using a z-score in this context?
4. How would you calculate the confidence interval if the population standard deviation was known?
5. Why is it necessary to assume the sample is random?

### Tip:
To reduce the margin of error and make the confidence interval narrower, consider increasing the sample size or lowering the standard deviation.

In a random sample of 49 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3441 with a standard deviation of $2533. Construct and interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns.

Calculate the 90% confidence interval for the mean additional tax owed from a sample of 49 audited estate tax returns. Learn step-by-step how to compute the margin of error and interpret the interval using the mean $3441 and standard deviation $2533.

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