Problem 28 - Basic College Mathematics

Question

Unemployment Rates. In August 2017 , the United States unemployment rate was \(4.4 \%\). The following list gives the unemployment rates for the 26 states with rates statistically different from the nation's rate. Create a stem-and- leaf plot to represent the data. $$ \begin{array}{l} 7.2,3.5,5.1,2.4,6.4,2.6,2.9,3.5,3.3,3.9,5.4,3.8 \\ 3.8,5.3,3.9,2.8,2.7,6.3,2.3,5.4,3.3,3.3,3.5,3.0 \end{array} $$ 3.8,3.4

Step-by-Step Solution

Answer

The stem-and-leaf plot is: 2 | 3 4 6 7 8 9, 3 | 0 3 3 3 4 5 5 5 8 8 8 9 9, 5 | 1 3 4 4, 6 | 3 4, 7 | 2.

Step 1: Organize the Data

Start by organizing the unemployment rates in ascending order to have a clear view of the data. The sorted data is: 2.3, 2.4, 2.6, 2.7, 2.8, 2.9, 3.0, 3.3, 3.3, 3.3, 3.4, 3.5, 3.5, 3.5, 3.8, 3.8, 3.8, 3.9, 3.9, 5.1, 5.3, 5.4, 5.4, 6.3, 6.4, 7.2.

Step 2: Identify the Stems

Identify the stems, which are the integer parts of the unemployment rates. The stems are: 2, 3, 5, 6, and 7.

Step 3: List the Leaves

For each stem, list the leaves, which are the decimal parts of the unemployment rates. For example, if the unemployment rate is 2.3, the stem is 2, and the leaf is 3.

Step 4: Create the Stem-and-Leaf Plot

Arrange the stems in a vertical column and list the leaves to the right of each stem. The stem-and-leaf plot will look like this:2 | 3 4 6 7 8 93 | 0 3 3 3 4 5 5 5 8 8 8 9 95 | 1 3 4 46 | 3 47 | 2

Key Concepts

Unemployment Rates

Unemployment rates are a key economic indicator. They show the percentage of people in the labor force who are jobless and actively seeking employment. In August 2017, the United States had an unemployment rate of 4.4%, marking a significant insight into the nation’s economic health. Understanding how different states compare to the national rate can help policymakers make informed decisions.

Data Organization

Effective data organization is crucial in statistics. For tasks such as creating a stem-and-leaf plot, the first step is to sort the data in ascending order. This allows us to see the distribution and identify patterns more easily. In our example, unemployment rates for 26 states are listed in ascending order, starting from 2.3 and ending at 7.2. Organizing data helps in creating clear and interpretable charts and graphs.

Statistical Representation

One of the simplest forms of statistical representation is the stem-and-leaf plot. This plot is a handy tool for visually displaying numerical data. In a stem-and-leaf plot, each data point is split into a 'stem' (the leading digit(s)) and a 'leaf' (the trailing digit). This display organizes data to show its distribution. For example, in the unemployment rates data, '2' can be a stem and '3, 4, 6, 7, 8, 9' the leaves. This simplification makes it easier to understand the spread and frequency of the data.

Descriptive Statistics

Descriptive statistics involves summarizing and organizing data so that it can be easily understood. When we use a stem-and-leaf plot, we create a clear, organized visual representation of our data. Other measures in descriptive statistics include mean, median, mode, and range. By organizing unemployment rates into a stem-and-leaf plot, we can quickly see how data is clustered, making it easier to identify trends and outliers. This type of visualization helps to describe the basic features of our data clearly and efficiently.