Chapter 3.1, Problem 8E

Middle-Level Manager Salaries. Suppose that an independent study of middle-level managers employed at companies located in Atlanta, Georgia, was conducted to compare the salaries of managers working at firms in Atlanta to the salaries of middle-level managers across the nation. The following data show the salary, in thousands of dollars, for a sample of 15 middle-level managers employed at companies in the Atlanta area.

1. a. Compute the median salary for the sample of 15 middle-level managers. Suppose the median salary of middle-level managers employed at companies located across the nation is $85,000. How does the median salary for middle-level managers in the Atlanta area compare to the median for managers across the nation?
2. b. Compute the mean annual salary for managers in the Atlanta area and discuss how and why it differs from the median computed in part (a) for Atlanta area managers.
3. c. Compute the first and third quartiles for the salaries of middle-level managers in the Atlanta area.

To determine

Find the median salary for the sample of 15 middle-level managers.

Compare the median obtained median with the median reported by Wall Street Journal.

Answer to Problem 8E

The median salary for the sample of 15 middle- level managers is $80,000.

Explanation of Solution

Calculation:

The data set represent the salary (in thousand dollars) for a sample of 15 middle-level managers.

Software Procedure:

Step by step procedure to obtain the descriptive statistics using EXCEL is as follows:

• In an EXCEL sheet enter 15 salaries and label it as Salary.
• Go to Data > Data Analysis (in case it is not default, take the Analysis ToolPak from Excel Add Ins) > Descriptive statistics.
• Enter Input Range as $A$2:$A$15, select Columns in Grouped By, tick on Summary statistics.
• Click on OK.

Output using EXCEL is given as follows:

From the EXCEL output, the median salary is 80.

Since, the salary is given in thousand dollars, the median salary for a sample of 15 level managers can be inferred as $80,000.

The Wall Street Journal reported that the median salary for middle-level manager jobs is approximately $85,000.

Here, $80,000 < $85,000.

Therefore, the median salary for the sample of 15 middle-level managers working at firms in City A is slightly lower than the median salary reported by the Wall Street Journal.

To determine

Find the mean annual salary.

Explain the reason behind the existence of difference between computed mean and median in part (a).

Answer to Problem 8E

The mean annual salary is $84,000.

The distribution of salaries for middle-level managers working at firms in City A is right skewed.

Explanation of Solution

From the excel output, it is clear that the mean annual salary is 84.

Here, the median salary is less than the mean salary for the sample of 15 middle- level managers.

Skewness:

In statistics, skewness is a measure by which anyone can measure the asymmetrical behaviour of a distribution.

A distribution is said to be symmetric, when its mean, median, and mode are the same. In other words it can be said that, the length of the curve of the left-hand tail will be a mirror image of the right-hand tail.

Therefore, the shape of a distribution can be identified by using the measure of skewness.

Left-skewed:

For a left-skewed distribution, $\text{Mean}<\text{Median}<\text{Mode}$. Moreover, the data will monotonically decrease after attaining the maximum and if the left hand side of the curve is larger than the right hand side, then it is called a left-skewed distribution. This is also called as negatively-skewed distribution.

Right-skewed:

For a right-skewed distribution, $\text{Mean > Median > Mode}$. Moreover, the data will monotonically increase after attaining the maximum and if the right hand side of the curve is larger than the left hand side, then it is called a right-skewed distribution. This is also called as positively-skewed distribution.

Since, mean is greater than median, it can be inferred that the distribution of annual salary is right or positive skewed.

To determine

Obtain the first and third quartiles of the dataset.

Answer to Problem 8E

The first and third quartiles of the dataset are $\underset{\_}{Q_1=67}$, and $\underset{\_}{Q_3=106}$, respectively.

Explanation of Solution

Calculation:

Quartiles:

Quartiles are those values which divide the data into four equal parts.

Quartiles tells that 25% of the values will be less than the first quartile and 25% of the values will be greater than the third quartile and 50% of the values lies between the first and third quartile with median dividing the parts.

The general formula to obtain the location of $p^{th}$ percentile is given below:

$L_p=\frac{p}{100}\left(n+1\right)$

First quartile:

The first quartile $Q_1$ of a dataset is a value that divides the lowest 25% of the observations and highest 75% percent of observations.

Therefore, the first quartile is the 25^th percentile.

The location of 25^th percentile is calculated as follows:

$\begin{array}{c}{L}_{25}=\frac{25}{100}\left(15+1\right)\\ =\frac{25\left(16\right)}{100}\\ =\frac{400}{100}\\ =4\end{array}$

Thus, the location of 25^th percentile is 4.0.

From this it can be said that, the 25^th percentile is 0% of the way between the value in 4^th position of ordered dataset and the value in 5^th position of ordered dataset.

The dataset of annual salary have to be arranged in ascending order.

The values corresponding to 4^th and 5^th positions of ordered dataset of annual salaries are 67 and 73.

The 25^th percentile is the value in 4^th position (67) plus 0.0 times the difference between the value in 4^th position and 5^th position.

The 25^th percentile is obtained as given below:

$\begin{array}{c}{\text{25}}^{\text{th}}\text{percentile}=67+0.0\left(73-67\right)\\ =67\end{array}$

Hence, the 25^th percentile or first quartile is 67.

Third quartile:

The third quartile $Q_3$ of a dataset is a value that divides the lowest 75% of the observations and highest 25% percent of observations.

Therefore, the third quartile is the 75^th percentile.

The location of 75^th percentile is calculated as follows:

$\begin{array}{c}{L}_{75}=\frac{75}{100}\left(15+1\right)\\ =\frac{75\left(16\right)}{100}\\ =\frac{1,200}{100}\\ =12\end{array}$

Thus, the location of 75^th percentile is 12.

From this it can be said that, the 75^th percentile is 0% of the way between the value in 12^th position of ordered dataset and the value in 13^th position of ordered dataset.

The dataset of annual salary have to be arranged in ascending order.

The values corresponding to 12^th and 13^th positions of ordered dataset of annual salaries are 106 and 108.

The 75^th percentile is the value in 12^th position (106) plus 0.0 times the difference between the value in 12^th position and 13^th position.

The 75^th percentile is obtained as given below:

$\begin{array}{c}{\text{75}}^{\text{th}}\text{percentile}=106+0.0\left(108-106\right)\\ =106\end{array}$

Hence, the 75^th percentile or third quartile is 106.

Thus, the first and third quartiles of the dataset are $\underset{\_}{Q_1=67}$, and $\underset{\_}{Q_3=106}$, respectively.