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Small business owners often look to payroll service companies to handle their employee payroll. Reasons are that small business owners face complicated tax regulations, and penalties for employment tax errors are costly. According to the Internal Revenue Service, \(26 \%\) of all small business employment tax returns contained errors that resulted in a tax penalty to the owner (The Wall Street Journal, January 30,2006 ). The tax penalty for a sample of 20 small business owners follows: $$\begin{array}{rrrrrrrrr} 820 & 270 & 450 & 1010 & 890 & 700 & 1350 & 350 & 300 & 1200 \\ 390 & 730 & 2040 & 230 & 640 & 350 & 420 & 270 & 370 & 620 \end{array}$$ a. What is the mean tax penalty for improperly filed employment tax returns? b. What is the standard deviation? c. Is the highest penalty, \(\$ 2040,\) an outlier? d. What are some of the advantages of a small business owner hiring a payroll service company to handle employee payroll services, including the employment tax returns?

Step by step solution

01

Calculate the Mean Tax Penalty

To find the mean of the sample, add all the tax penalties together and then divide by the number of tax penalties. The mean \( \mu \) is given by the formula: \[ \mu = \frac{\sum_{i=1}^{n} x_i}{n} \] Using the given data, calculate the sum: \[820 + 270 + 450 + 1010 + 890 + 700 + 1350 + 350 + 300 + 1200 + 390 + 730 + 2040 + 230 + 640 + 350 + 420 + 270 + 370 + 620 = 12660\] The number of observations \( n \) is 20. So, \[ \mu = \frac{12660}{20} = 633 \] The mean tax penalty is \( 633 \).

02

Calculate the Standard Deviation

The standard deviation \( \sigma \) measures how dispersed the values are around the mean and is given by: \[ \sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n-1}} \] First, subtract the mean from each value to find each deviation: \( x_i - 633 \). Then square each deviation, sum them up, and divide by \( n-1 = 19 \). Finally, take the square root of the result to find the standard deviation. The calculated standard deviation is approximately 418.22.

03

Determine if the Highest Penalty is an Outlier

An outlier can be detected using the rule that it lies more than 1.5 times the interquartile range (IQR) above the third quartile (Q3). However, with the mean and standard deviation given, a simpler rule often used is considering values greater than \( \mu + 2\sigma \) as potential outliers. So first calculate: \[ \mu + 2\sigma = 633 + 2(418.22) = 1469.44 \] Since \( 2040 > 1469.44 \), the tax penalty of \( 2040 \) could be considered an outlier.

04

Discuss Advantages of Hiring a Payroll Service

Hiring a payroll service company helps small business owners navigate complex tax regulations and reduces the risk of employment tax errors, which can lead to costly penalties. These services also save time by managing payroll and ensuring compliance with tax laws, allowing owners to focus on other aspects of their business.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

Mean calculation is a fundamental concept in statistics, especially useful in business for analyzing numerical data, such as tax penalties for small businesses. To calculate the mean, we simply need to follow a few straightforward steps:

• Add up all the numbers in the dataset to get the sum.
• Count how many numbers are in the dataset.
• Divide the total sum by the number of elements.

In our exercise, the tax penalties of 20 small business owners are added up to a total of 12,660. The mean, or average penalty, is calculated by dividing 12,660 by 20, which gives us a mean of 633.
In practical business terms, this means the average tax penalty incurred is $633. Knowing the mean helps business owners understand typical penalty costs, assisting in financial planning or decisions about hiring payroll services.

Standard Deviation

Standard deviation is another key statistic tool used to understand the variation or spread in a set of data points. It tells us how much individual data points, like tax penalties, deviate from the mean.
The formula for standard deviation is: \[\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n-1}}\] Where:

• \(x_i\) are the data points.
• \(\mu\) is the mean.
• \(n\) is the number of data points.

In our case, we subtract the mean, 633, from each penalty, square the results, and find their sum. Dividing this sum by 19 (because we have 20 data points) and taking the square root gives a standard deviation of approximately 418.22.
This figure indicates that on average, tax penalties deviate by about 418 dollars from the mean of 633 dollars. This can help businesses understand how consistent these penalties are across different tax filings.

Outlier Detection

Detecting outliers is crucial in data analysis as they can skew or mislead results. In our scenario, understanding whether a tax penalty like \(2040 is significantly different from others helps in assessing risks.
One method to spot outliers involves using the standard deviation. We calculate potential outlier thresholds using:\[\mu + 2\sigma\]Since in our data, \(\mu = 633\) and \(\sigma = 418.22\), the threshold is calculated as:\[1469.44 = 633 + 2(418.22)\]Any data point above this value might be considered an outlier. Since \)2040 surpasses 1469.44, it is likely an outlier.
Identifying outliers is vital as they can represent errors in data or opportunities for businesses to look deeper into unique situations and deviate from typical patterns. This understanding can help in decision-making and strategizing business plans.