Americas
Europe
133E
Expert-verifiedBest-paid CEOs. Refer to Glassdoor Economic Research firm’s 2015 ranking of the 40 best-paid CEOs in Table 2.1 (p. 65). Recall that data were collected on a CEO’s current salary, age, and the ratio of salary to a typical worker’s pay at the firm.
a. Create a scatterplot to relate a CEO’s ratio of salary to worker pay to the CEO’s age. Comment on the strength of the association between these two variables.
b. Conduct an outlier analysis of the ratio variable. Identify the highly suspect outlier in the data.
c. Remove the highly suspect outlier from the data and recreate the scatterplot of part a. What do you observe?
a. The graph is given below:
No strong association
b.Highly suspect outlier = 1951
c. No major change
The graph is given below:
There is no visible strong association between the CEO’s age and the ratio of the CEO’s salary to that of a typical worker. The data is clustered to the left of the graph except for a few scattered points.
We will use the box plot method to conduct an outlier analysis on the ratio variable.
We will first calculate the quartiles, IQR, Inner, and Outer fences.
Arranging the data in ascending order,
$\begin{array}{c}\text{Q1=}\frac{\text{N+1}}{\text{4}}\\ =\frac{40+1}{4}\\ =\frac{41}{4}\\ {\text{=10.25}}^{\text{th}}\text{}\end{array}$
$\text{Term = 483}$
$\begin{array}{c}\text{Q2=}\frac{\text{N+1}}{\text{2}}\\ =\frac{40+1}{2}\\ =\frac{41}{2}\\ {\text{= 20.25}}^{\text{th}}\text{}\end{array}$
$\text{Term = 536.5}$
$\begin{array}{c}\text{Q3=}\frac{\text{3}\left(\text{N+1}\right)}{\text{4}}\\ \text{=}\frac{3\left(40+1\right)}{4}\\ =\frac{123}{4}\\ {\text{= 30.75}}^{\text{th}}\text{}\end{array}$
$\text{Term = 644.25}$
$\begin{array}{c}{\text{IQR = Q}}_{\text{U}}{\text{\u2013 Q}}_{\text{L}}\text{}\\ \text{= 644.25 \u2013 483}\\ \text{= 161.25}\end{array}$
$\begin{array}{c}\text{Lower Inner Fence}={\text{Q}}_{\text{L}}-\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{483}\u2013\text{1}.\text{5}\left(\text{161}.\text{25}\right)\\ =\text{483}\u2013\text{241}.\text{875}\\ =\text{241}.\text{125}\end{array}$
$\begin{array}{c}\text{Upper Inner Fence}={\text{Q}}_{\text{U}}+\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{644}.\text{25}+\text{1}.\text{5}\left(\text{161}.\text{25}\right)\\ =\text{644}.\text{25}+\text{241}.\text{875}\\ =\text{886}.\text{125}\end{array}$
Now we will plot these,
The graph shows that the outliers are 1951, 1522, 1192, 1133, and 939.
To find which of these are highly suspect outliers (with a z-score > 3), we will check the z-score of these outliers.
First, we will calculate the mean and standard deviation of the data.
$\begin{array}{c}\text{Mean =}\frac{\text{Sum of all observations}}{\text{No. of observations}}\\ \text{=}\frac{25670}{40}\\ =\text{}641.75\end{array}$
$\begin{array}{c}\text{Variance =}\frac{\sum {\left(\text{\chi}-\right)}^{2}}{\text{n}-\text{1}}\\ \text{=}\frac{\text{3864170}}{\text{39}}\\ \text{= 99081.28}\end{array}$
$\begin{array}{c}\text{Standard Deviation =}\sqrt{\text{Variance}}\\ =\sqrt{99081.28}\\ =314.77\end{array}$
Mean = 641.75 and Standard Deviation = 314.77
$\begin{array}{c}\text{z-score of 1951=}\frac{\text{1951}-\text{641.75}}{\text{314.77}}\\ \text{=}\frac{\text{1309.25}}{\text{314.77}}\\ \text{=4.159}\end{array}$
$\begin{array}{c}\text{z-score of 1522 =}\frac{\text{1522}-\text{641.75}}{\text{314.77}}\\ \text{=}\frac{\text{880.25}}{\text{314.77}}\\ \text{= 2.79}\end{array}$
Based on the values of the z-score, 1522 is not a highly suspect outlier. Therefore, none of the values lower than 1522 are outliers.
So, the only highly suspect outlier is 1951.
1951 is a highly suspect outlier. Therefore, we will remove that from the data and create the scatterplot.
There is not much difference in the scatterplot after removing the highly suspect outlier.
Top credit card issuers, by region. The Nilson Report (December 2015) published a list of the top 150 credit card issuers worldwide. The issuers (e.g., American Express, MasterCard, Visa) were ranked based on outstanding debt during the year. The table gives a breakdown of the regions in the world served by the top 150 credit card issuers.
Worldwide Region | Number of credit card issuers | |
Asia-Pacific | 48 | |
Canada | 10 | |
Europe | 34 | |
Latin America | 29 | |
Middle East/Africa | 3 | |
United States | 26 | |
Total | 150 |
a. One of the top 150 credit card issuers is selected at random, and the region it serves is determined. What type of data (quantitative or qualitative) is measured?
b. For each region in the table, calculate the percentage of the 150 top credit card issuers that fall into that region.
c. Use the percentages, part b, to construct a relative frequency bar graph for the data summarized in the table.
d. Based on the bar graph, make a statement about the regions that most of the top 150 credit card users serve.
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