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Chapter 1: Problem 86

Last year a small accounting firm paid each of its five clerks \(\$ 22,000\), two junior accountants \(\$ 50,000\) each, and the firm's owner \(\$ 270,000 .\) What is the mean salary paid at this firm? How many of the employees earn less than the mean? What is the median salary? Write a sentence to describe how an unethical recruiter could use statistics to mislead prospective employees.

Short Answer

Expert verified
Mean salary is $60,000; 7 employees earn less than the mean; the median salary is $22,000.

Step by step solution

01

Identify Salaries and Employee Count

First, list each salary and determine the number of employees earning each one: - Five clerks earn $22,000 each. - Two junior accountants earn $50,000 each. - One owner earns $270,000. This aligns with: - 5 individuals earning $22,000 - 2 individuals earning $50,000 - 1 individual earning $270,000 The total number of employees is 5 + 2 + 1 = 8.
02

Calculate Total Salaries

Calculate the total salary expenditure for the firm:- Total for clerks: \(5 \times 22,000 = 110,000\)- Total for junior accountants: \(2 \times 50,000 = 100,000\)- Owner's salary: \(270,000\)Add all these:\[ 110,000 + 100,000 + 270,000 = 480,000 \]
03

Calculate the Mean Salary

Find the mean salary by dividing the total salaries by the number of employees:\[ \text{Mean Salary} = \frac{480,000}{8} = 60,000 \]
04

Determine Employees Earning Less than the Mean

Identify who earns less than the mean salary of $60,000: - Five clerks earn $22,000 - Two junior accountants earn $50,000 Thus, 5 (clerks) + 2 (junior accountants) = 7 employees earn less than the mean.
05

Calculate the Median Salary

The median salary is the middle value of the ordered salaries, which requires us to list the salaries in order:- \(22,000, \)22,000, \(22,000, \)22,000, \(22,000, \)50,000, \(50,000, \)270,000With 8 numbers, the median is the average of the 4th and 5th terms:\[ \text{Median} = \frac{22,000 + 22,000}{2} = 22,000 \]
06

Describe Misleading Use of Statistics

An unethical recruiter might emphasize the mean salary of $60,000 to suggest higher average pay, while the median salary of $22,000 shows a lower usual pay rate typical employees receive. This may mislead candidates about potential earnings.

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Key Concepts

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

Mean Salary Calculation
The mean salary is a key statistic that provides an overview of the average earnings in a company. To calculate the mean salary, you need to find the total sum of all salaries and divide that by the number of employees. In this case, the firm has a total salary expenditure of \(480,000\) and employs 8 people. So, the mean salary is computed as follows:
\[ \text{Mean Salary} = \frac{480,000}{8} = 60,000 \]
This calculation shows that the average salary per employee is \(60,000. The mean is sensitive to extreme values, such as the owner's \\)270,000 salary, which significantly raises the average.
Median Salary
Unlike the mean, the median salary provides a better indication of what a typical employee might earn, especially when salary distributions are skewed. The median is the middle value in an ordered list of numbers. In this firm's case, the ordered salaries are:
\(22,000, 22,000, 22,000, 22,000, 22,000, 50,000, 50,000, 270,000\).
Since there are 8 salaries, the median is the average of the 4th and 5th values:
\[ \text{Median} = \frac{22,000 + 22,000}{2} = 22,000 \]
Thus, the median salary is $22,000, reflecting the earnings of most employees more accurately than the mean.
Misuse of Statistics
Statistics can sometimes be used to mislead, particularly if one understands how to present data selectively. An unethical recruiter might highlight the mean salary of \\(60,000 to entice prospective employees by suggesting that they could earn a high salary. However, without context, this mean salary can be deceptive because it is skewed by the owner's substantial earnings. It does not represent the majority of employees' experiences, who earn significantly less, as indicated by the median salary of \\)22,000. Therefore, potential employees should look at both mean and median salaries to understand the true earning potential within a firm.
Statistics in Business Context
In a business setting, understanding measures of central tendency like mean and median is crucial for evaluating pay structures and making informed decisions. Mean salary reveals average earnings, useful for assessing overall salary expenditure. However, it's critical to recognize when the mean might be distorted by outliers, like a highly-paid executive's salary.
  • The mean can give a false impression of general employee salaries if not viewed critically.
  • The median offers a clearer picture of the typical salary, particularly in skewed distributions.

Hence, using both mean and median provides a complete and accurate understanding of salary distributions, helping businesses to communicate honestly and plan effectively.

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Most popular questions from this chapter

The following bar graph shows the distribution of favorite subject for a sample of 1000 students. What is the most serious problem with the graph? (a) The subjects are not listed in the correct order. (b) This distribution should be displayed with a pie chart. (c) The vertical axis should show the percent of students. (d) The vertical axis should start at 0 rather than 100 . (e) The foreign language bar should be broken up by language.

Multiple choice: If a distribution is skewed to the right with no outliers, (a) mean \(<\) median. (b) mean \(\approx\) median. (c) mean = median. (d) mean \(>\) median. (e) We can't tell without examining the data.

Multiple choice: The scores on a statistics test had a mean of 81 and a standard deviation of \(9 .\) One student was absent on the test day, and his score wasn't included in the calculation. If his score of 84 was added to the distribution of scores, what would happen to the mean and standard deviation? (a) Mean will increase, and standard deviation will increase. (b) Mean will increase, and standard deviation will decrease. (c) Mean will increase, and standard deviation will stay the same. (d) Mean will decrease, and standard deviation will increase. (e) Mean will decrease, and standard deviation will decrease.

Which of the following is the best reason for choosing a stemplot rather than a histogram to display the distribution of a quantitative variable? (a) Stemplots allow you to split stems; histograms don't. (b) Stemplots allow you to see the values of individual observations. (c) Stemplots are better for displaying very large sets of data. (d) Stemplots never require rounding of values. (e) Stemplots make it easier to determine the shape of a distribution.

Here are the amounts of money (cents) in coins carried by 10 students in a statistics class: 50,35,0,97 , \(76,0,0,87,23,65 .\) To make a stemplot of these data, you would use stems (a) 0,1,2,3,4,5,6,7,8,9 (b) 0,2,3,5,6,7,8,9 (c) 0,3,5,6,7 (d) 00,10,20,30,40,50,60,70,80,90 (e) None of these.
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