If 49 Taxpayers Are Randomly Selected, What Is The Probability That, On Average, They Took More Than 9.87 Hours To Complete Form 1040? To Answer This Question, Complete The Following Steps:1. Let The Random Variable X ˉ \bar{X} X ˉ Be The Average

Introduction

The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the behavior of large sample means. It states that, given certain conditions, the distribution of the sample mean will be approximately normal, even if the underlying population distribution is not normal. In this article, we will explore the application of the CLT in a real-world scenario, specifically in the context of tax return processing.

Step 1: Define the Random Variable

Let the random variable $\bar{X}$ be the average time taken by 49 taxpayers to complete Form 1040. We are interested in finding the probability that $\bar{X}$ is greater than 9.87 hours.

Step 2: Determine the Population Distribution

The population distribution of the time taken to complete Form 1040 is not explicitly stated. However, we can assume that it is a continuous random variable with a mean $\mu$ and standard deviation $\sigma$. We will use the CLT to approximate the distribution of $\bar{X}$.

Step 3: Apply the Central Limit Theorem

The CLT states that, given certain conditions, the distribution of the sample mean $\bar{X}$ will be approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$, where $n$ is the sample size. In this case, $n = 49$. Therefore, the distribution of $\bar{X}$ will be approximately normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{49}} = \frac{\sigma}{7}$.

Step 4: Find the Probability

We want to find the probability that $\bar{X}$ is greater than 9.87 hours. This can be written as $P(\bar{X} > 9.87)$. Using the CLT, we can standardize this probability by converting it to a z-score:

$$z = \frac{\bar{X} - \mu}{\frac{\sigma}{7}} = \frac{9.87 - \mu}{\frac{\sigma}{7}}$$

We can then use a standard normal distribution table or calculator to find the probability that $z > z_{\text{critical}}$, where $z_{\text{critical}}$ is the z-score corresponding to the desired probability.

Step 5: Calculate the Z-Score

To calculate the z-score, we need to know the values of $\mu$ and $\sigma$. Unfortunately, these values are not provided. However, we can assume that the population distribution is approximately normal with a mean $\mu = 8$ hours and a standard deviation $\sigma = 2$ hours. This is a reasonable assumption, as the time taken to complete Form 1040 is likely to be normally distributed.

Using these values, we can calculate the z-score:

$$z = \frac{9.87 - 8}{\frac{2}{7}} = \frac{1.87}{\frac{2}{7}} = \frac{1.87 \times 7}{2} = 6.545$$

Step 6: Find the Probability

Using a standard normal distribution table or calculator, we can find the probability that $z > 6.545$. This is approximately equal to 0.

Conclusion

In conclusion, the probability that, on average, 49 taxpayers took more than 9.87 hours to complete Form 1040 is approximately 0. This is because the z-score is extremely large, indicating that the probability is very close to 0.

Limitations

This analysis has several limitations. Firstly, the population distribution of the time taken to complete Form 1040 is not explicitly stated. Secondly, the values of $\mu$ and $\sigma$ are assumed, rather than being known. Finally, the CLT assumes that the sample size is large, which may not be the case in this scenario.

Future Research Directions

Future research directions could include:

• Collecting data on the time taken to complete Form 1040 for a large sample of taxpayers
• Analyzing the distribution of the time taken to complete Form 1040 using statistical methods
• Developing more accurate models for predicting the time taken to complete Form 1040

References

• Central Limit Theorem. (n.d.). In Encyclopedia of Mathematics. Retrieved from https://encyclopediaofmath.org/index.php/Central_Limit_Theorem
• Johnson, R. A., & Bhattacharyya, G. K. (2014). Statistics: Principles and Methods. John Wiley & Sons.
• Moore, D. S., & McCabe, G. P. (2013). Introduction to the Practice of Statistics. W.H. Freeman and Company.
  Frequently Asked Questions (FAQs) about the Central Limit Theorem and Tax Return Processing =====================================================================================

Q: What is the Central Limit Theorem (CLT)?

A: The Central Limit Theorem is a fundamental concept in statistics that describes the behavior of large sample means. It states that, given certain conditions, the distribution of the sample mean will be approximately normal, even if the underlying population distribution is not normal.

Q: How does the CLT apply to tax return processing?

A: The CLT can be applied to tax return processing by assuming that the time taken to complete Form 1040 is a continuous random variable with a mean and standard deviation. The CLT can then be used to approximate the distribution of the sample mean time taken to complete Form 1040 for a large sample of taxpayers.

Q: What are the assumptions of the CLT?

A: The CLT assumes that the sample size is large, the population distribution is continuous, and the observations are independent and identically distributed.

Q: What is the significance of the CLT in tax return processing?

A: The CLT is significant in tax return processing because it allows us to make inferences about the population distribution of the time taken to complete Form 1040 based on a sample of taxpayers. This can be useful for tax authorities in setting deadlines for tax returns and for taxpayers in planning their tax preparation.

Q: How can the CLT be used to improve tax return processing?

A: The CLT can be used to improve tax return processing by:

• Identifying the most time-consuming steps in the tax preparation process
• Developing more accurate models for predicting the time taken to complete Form 1040
• Improving the efficiency of tax preparation software
• Providing more accurate estimates of the time taken to complete Form 1040 for taxpayers

Q: What are the limitations of the CLT in tax return processing?

A: The CLT has several limitations in tax return processing, including:

• The assumption of a large sample size may not be met in practice
• The population distribution of the time taken to complete Form 1040 may not be continuous
• The observations may not be independent and identically distributed
• The CLT may not be able to capture the complexity of the tax preparation process

Q: How can the CLT be extended to other areas of tax return processing?

A: The CLT can be extended to other areas of tax return processing by:

• Applying the CLT to other types of tax returns, such as corporate tax returns
• Using the CLT to analyze the distribution of other variables, such as tax liability or audit rates
• Developing more complex models that incorporate multiple variables and interactions

Q: What are the future research directions for the CLT in tax return processing?

A: Future research directions for the CLT in tax return processing include:

• Collecting more data on the time taken to complete Form 1040 for a large sample of taxpayers
• Developing more accurate models for predicting the time taken to complete Form 1040
• Analyzing the distribution of other variables, such as tax liability or audit rates
• Developing more complex models that incorporate multiple variables and interactions

Q: How can the CLT be used to improve tax policy?

A: The CLT can be used to improve tax policy by:

• Providing more accurate estimates of the time taken to complete Form 1040 for taxpayers
• Identifying the most time-consuming steps in the tax preparation process
• Developing more accurate models for predicting the time taken to complete Form 1040
• Improving the efficiency of tax preparation software

Q: What are the implications of the CLT for tax authorities?

A: The CLT has several implications for tax authorities, including:

• The need to collect more data on the time taken to complete Form 1040 for a large sample of taxpayers
• The need to develop more accurate models for predicting the time taken to complete Form 1040
• The need to improve the efficiency of tax preparation software
• The need to provide more accurate estimates of the time taken to complete Form 1040 for taxpayers