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Part 1

In a random sample of 

100100 

audited estate tax​ returns, it was determined that the mean amount of additional tax owed was 

​$34753475 

with a standard deviation of 

​$25072507. 

Construct and interpret a​ 90% confidence interval for the mean additional amount of tax owed for estate tax returns.

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Part 1

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.

One can be​ 90% confident that the mean additional tax owed is between 

​$enter your response here 

and 

​$enter your response here.

B.

​90% of taxes owed for estate tax returns are between 

​$enter your response here 

and 

​$enter your response here.

C.

There is a​ 90% probability that the mean additional tax owed is between 

​$enter your response here 

and 

​$enter your response here.

To construct a 90% confidence interval for the mean additional tax owed for estate tax returns, we can use the following formula for the confidence interval (CI):

\[
\text{CI} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right)
\]

Where:
- $\bar{x} = 34753475$ (sample mean)
- $s = 25072507$ (sample standard deviation)
- $n = 100$ (sample size)
- $z^*$ is the z-score corresponding to the desired confidence level.

### Step 1: Find the z-score for a 90% confidence level.
The z-score for a 90% confidence level (using a z-table or calculator) is approximately $z^* = 1.645$.

### Step 2: Calculate the standard error (SE).
\[
\text{SE} = \frac{s}{\sqrt{n}} = \frac{25072507}{\sqrt{100}} = \frac{25072507}{10} = 2507250.7
\]

### Step 3: Calculate the margin of error (ME).
\[
\text{ME} = z^* \cdot \text{SE} = 1.645 \cdot 2507250.7 \approx 4123937.45
\]

### Step 4: Construct the confidence interval.
\[
\text{Lower limit} = \bar{x} - \text{ME} = 34753475 - 4123937.45 \approx 30629537.55
\]
\[
\text{Upper limit} = \bar{x} + \text{ME} = 34753475 + 4123937.45 \approx 38877412.45
\]

### Final Confidence Interval
Rounding to the nearest dollar:
- Lower limit: $30,629,538
- Upper limit: $38,877,412

### Interpretation
The correct interpretation is:
**A. One can be 90% confident that the mean additional tax owed is between $30,629,538 and $38,877,412.**

### Summary
The 90% confidence interval for the mean additional amount of tax owed for estate tax returns is $\$30,629,538$ to $\$38,877,412$.

Do you want details on any specific part or have any questions? Here are some follow-up questions you might consider:

1. How would the confidence interval change with a larger sample size?
2. What are the implications of the standard deviation in this context?
3. How do you interpret the concept of confidence intervals in general?
4. What factors could affect the accuracy of this confidence interval?
5. How would you approach constructing a confidence interval for a different confidence level (e.g., 95%)? 

**Tip:** Always check the underlying assumptions of normality and sample size when constructing confidence intervals.

In a random sample of 100 audited estate tax returns, it was determined that the mean amount of additional tax owed was $34,753,475 with a standard deviation of $25,072,507. Construct and interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns.

Explore how to construct a 90% confidence interval for the mean additional amount of tax owed for estate tax returns, based on a sample of 100 returns with a mean of $34,753,475 and standard deviation of $25,072,507. This analysis provides a range between $30,629,538 and $38,877,412.

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