Problem 30 - Basic Statistics for Business and Economics with Student

Question

In 1996 a total of 14,968,000 taxpayers in the United States filed their individual tax returns electronically. By the year 2006 the number increased to 73,300,000 . What is the geometric mean annual increase for the period?

Step-by-Step Solution

Answer

The geometric mean annual increase is 17.8%.

Step 1: Understand the Problem

We need to calculate the geometric mean annual increase in the number of taxpayers filing electronically from 1996 to 2006. The data given includes the number of taxpayers in both 1996 and 2006.

Step 2: Identify the Formula for Geometric Mean

The geometric mean annual increase can be determined by the formula \( r = \left(\frac{V_f}{V_i}\right)^{\frac{1}{n}} - 1 \), where \( V_f \) is the final value, \( V_i \) is the initial value, and \( n \) is the number of years.

Step 3: Substitute the Given Values into the Formula

Here, \( V_f = 73,300,000 \), \( V_i = 14,968,000 \), and \( n = 10 \) years (from 1996 to 2006). Substitute these values into the formula: \[ r = \left(\frac{73,300,000}{14,968,000}\right)^{\frac{1}{10}} - 1 \]

Step 4: Calculate the Ratio

Divide the final value by the initial value to find the ratio: \[ \frac{73,300,000}{14,968,000} = 4.8954 \]

Step 5: Calculate the 10th Root

Calculate the 10th root of the ratio found in Step 4: \[ 4.8954^{\frac{1}{10}} \approx 1.178 \]

Step 6: Get the Geometric Mean Annual Increase

Subtract 1 from the value found in Step 5 to get the geometric annual increase: \[ 1.178 - 1 = 0.178 \]

Step 7: Convert Decimal to Percentage

Convert the decimal to a percentage to express the annual increase rate as a percentage: \[ 0.178 \times 100\% = 17.8\% \]

Key Concepts

Annual Increase

To understand the concept of annual increase, especially in the context of percentage growth, it's crucial to divide the growth process over a specific number of years. Here, we are looking at the growth in the number of taxpayers using electronic filing over a decade. Annual increase helps us gauge how consistently a value grows each year over a given period.
The geometric mean annual increase is used because it takes into account the compounding effect over multiple periods. Unlike the arithmetic mean, which is more appropriate for consistent or linear growth, the geometric mean is ideal for exponential growth patterns.

• The starting value in 1996 was 14,968,000.
• The ending value in 2006 was 73,300,000.
• The period is 10 years.

By calculating this average growth each year, we find that the number of electronic filers increased by 17.8% annually. This means the number of taxpayers not only increased each year but the growth also compounded over the period, leading to a significant increase by the end of 2006.

Taxpayers

Taxpayers are individuals or entities obligated to file tax returns to the government. In the United States, millions of taxpayers file annually, and technological advancements have increasingly facilitated the filing process. Here, the focus is on those individuals who opted to file electronically.
Between 1996 and 2006, there was significant growth in the number of taxpayers adopting electronic filing. This shift can be attributed to several factors:

• Increased accessibility and ease of use of electronic filing systems.
• Government incentives for electronic filing, such as faster processing times.
• Broad acceptance and trust in digital processes.

This decade-long reviewed increase underscores the broader trend of digitization and the adaptation of technology in government services.

Electronic Filing

Electronic filing, often referred to as e-filing, is a method of submitting tax returns via the internet. By 1996, the idea of filing taxes online was just beginning to gain traction, and it has since become the norm for many taxpayers.
Advantages of electronic filing include:

• Increased accuracy in calculations as software automatically checks for errors.
• Significant reduction in processing time compared to paper filing.
• Convenient e-verification methods eliminate the need for paper signatures.
• Immediate confirmation upon successful filing.
• Security and privacy maintained through encryption and authentication technologies.

The rise in taxpayers using electronic filing from 14,968,000 in 1996 to 73,300,000 in 2006 highlights the growing trust and reliance on digital methods to meet tax obligations.

Mathematical Formula

The mathematical formula used to calculate the geometric mean annual increase is integral to understanding exponential growth over time. In this problem, the formula applied is:
\[ r = \left(\frac{V_f}{V_i}\right)^{\frac{1}{n}} - 1 \]Where:

• \( V_f \) is the final value of taxpayers in 2006, which is 73,300,000.
• \( V_i \) is the initial value of taxpayers in 1996, which is 14,968,000.
• \( n \) is the number of years from the initial to the final value, which is 10 years.

First, we determine the ratio of final to initial value, which is a growth factor over the entire period. Next, by taking the 10th root (because there are 10 years), we find the average factor of increase per year. Subtracting 1 converts this factor into a growth rate, showing an increase percentage over that time.
In essence, using this mathematical approach not only reveals the total increase but illuminates the yearly compounded growth rate, equipping us with a more nuanced understanding of growth dynamics.