91Ó°ÊÓ

The following data represent salaries, in thousands of dollars, for employees of a small company. Notice that the data have been sorted in increasing order. \(\begin{array}{lllllllllllll}24 & 25 & 25 & 27 & 27 & 29 & 30 & 35 & 35 & 35 & 36 & 38 & 38 \\ 39 & 39 & 40 & 40 & 40 & 45 & 45 & 45 & 45 & 47 & 52 & 52 & 52 \\ 58 & 59 & 59 & 61 & 61 & 67 & 68 & 68 & 68 & 250 & & & \end{array}\) (a) Make a histogram using the class boundaries \(23.5,69.5,115.5,161.5\), \(207.5,253.5\) (b) Look at the last data value. Does it appear to be an outlier? Could this be the owner's salary? (c) Eliminate the high salary of 250 thousand dollars. Make a new histogram using the class boundaries \(23.5,32.5,41.5,50.5,59.5,68.5 .\) Does this histogram reflect the salary distribution of most of the employees better than the histogram in part (a)? For Problems \(11-16\), use the specified number of classes to do the following. (a) Find the class width. (b) Make a frequency table showing class limits, class boundaries, midpoints, frequencies, relative frequencies, and cumulative frequencies. (c) Draw a histogram. (d) Draw a relative-frequency histogram. (e) Categorize the basic distribution shape as uniform, mound-shaped symmetrical, bimodal, skewed left, or skewed right. (f) Draw an ogive.

Step by step solution

01

Making a Histogram (Part a)

To create a histogram using the class boundaries \(23.5,69.5,115.5,161.5,207.5,253.5\), calculate the frequency of the salaries falling within each interval.- Class 1 (23.5 to 69.5): Include all but the last salary (250). There are 34 salaries in this range.- Class 2 (69.5 to 115.5): No salaries fall into this range.- Class 3 (115.5 to 161.5): No salaries fall into this range.- Class 4 (161.5 to 207.5): No salaries fall into this range.- Class 5 (207.5 to 253.5): One salary falls into this range (250).Now, draw a histogram with these frequencies.

02

Determine the Outlier Status (Part b)

Consider the last value, 250, in the data. An outlier is a data point that significantly differs from the rest of the dataset. Most of the salaries cluster around lower values, with 250 being extremely high compared to them. Thus, 250 is an outlier. It's plausible this could be the owner's salary, as it doesn't reflect the general salary trend shown by other data points.

03

Create New Histogram Without Outlier (Part c)

Remove the outlier, 250, and use the new class boundaries \(23.5,32.5,41.5,50.5,59.5,68.5\) to create a revised histogram:- Class 1 (23.5 to 32.5): 6 salaries (24 to 30).- Class 2 (32.5 to 41.5): 14 salaries (35 to 40).- Class 3 (41.5 to 50.5): 7 salaries (45 to 47).- Class 4 (50.5 to 59.5): 7 salaries (52 to 59).- Class 5 (59.5 to 68.5): 5 salaries (61 to 68).This histogram more accurately reflects the salary distribution of most employees.

04

Find Class Width

Using the specified class boundaries: - Class Width = Upper boundary of a class - Upper boundary of the previous class = 32.5 - 23.5 = 9 Each class in the specified frequency distribution should have a width of 9.

05

Create a Frequency Table

For the revised histogram without the outlier, populate a frequency table: | Class Limits | Class Boundaries | Midpoints | Frequency | Relative Frequency | Cumulative Frequency | |--------------|------------------|-----------|-----------|--------------------|----------------------| | 24-32 | 23.5-32.5 | 28 | 6 | 0.18 | 6 | | 33-41 | 32.5-41.5 | 37 | 14 | 0.41 | 20 | | 42-50 | 41.5-50.5 | 46 | 7 | 0.21 | 27 | | 51-59 | 50.5-59.5 | 55 | 7 | 0.21 | 34 | | 60-68 | 59.5-68.5 | 64 | 5 | 0.15 | 39 |

06

Draw Histograms

Draw two histograms: - One with absolute frequencies using the frequency table from Step 5. - One with relative frequencies. Scale the y-axis accordingly for relative frequencies.

07

Determine Distribution Shape

The histogram is mound-shaped and symmetrical because the bars form a bell-like shape, peaking in the middle and tapering off symmetrically on either side.

08

Draw Ogive

An ogive is a line graph of the cumulative frequency. Plot cumulative frequencies on the y-axis against the class boundaries on the x-axis from the frequency table. Start plotting from the lowest boundary, connecting points corresponding to cumulative frequencies.

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Key Concepts

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

Histogram

A histogram is a powerful tool for visualizing the distribution of data. It is essentially a bar chart that represents the frequency or relative frequency of data within specified intervals, known as classes or bins. When constructing a histogram, the x-axis displays these class intervals, while the y-axis shows the frequency or count of data points in each class.
To make a histogram, it's important first to choose appropriate class boundaries. This exercise uses specific boundaries such as 23.5, 32.5, 41.5, and so on, to group the salary data. Each salary falls into one of these ranges. The resulting histogram gives a clear visual representation of how many employees fall within each salary range.
Histograms help quickly identify patterns, central tendencies, and spread of the data. They can also highlight the presence of outliers or any unusual gaps and clusters, providing an insightful look into the dataset's structure.

Outlier

In statistics, an outlier is a data point that significantly deviates from other observations in the data set. It is either much higher or much lower than the rest of the values.
The salary data set includes a $250,000 entry, which is strikingly higher than the others, leading to its classification as an outlier. This single value can skew the results and may not accurately reflect the typical salary of the company's employees.
Outliers can arise from measurement errors, data entry errors, or they might represent a different population altogether. In some cases, such as executive salaries in a company, outliers can be real and provide meaningful insights, like indicating the presence of high earners such as company owners or business leaders. Removing outliers can sometimes make the data easier to analyze and represent the majority more accurately.

Frequency Table

A frequency table is a vital tool in organizing raw data into a concise and easy-to-read format. It displays how often each data range (or class) occurs within the dataset.
In our exercise, the frequency table divides salary data into different classes based on predefined boundaries. Each class has a corresponding frequency, indicating the number of salaries that fall within that range. When constructing a frequency table:

• List each class in one column.
• Record the frequency of each class in the adjacent column.
• Calculate relative frequencies, which show the proportion of the total count for each class.
• Determine the cumulative frequency, summing all frequencies up to the current class.

Creating a clear frequency table assists in understanding how the data is distributed across different ranges and helps in constructing histograms and analyzing distribution more effectively.

Distribution Shape

The distribution shape of a dataset describes how data points are spread across the range of values. Understanding the distribution's shape is critical for interpreting statistical results and knowing which statistical methods to apply.
One common shape is mound-shaped and symmetrical, resembling a bell curve, where most data points cluster around the central mode, tapering off equally on both sides. This is typical of a normal distribution.
Another shape could be skewed, either to the left or right, indicating that data is not evenly distributed. For instance, a right-skewed distribution means there are more low values with a few high outliers, as initially seen in the company salary data due to the $250,000 outlier.
Identifying the shape helps in understanding the data's central tendency and variability and in making more informed decisions when applying statistical tests.