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Chapter 2: Problem 27

A report from Texas Transportation Institute (Texas A\&M University System, 2005 ) titled "Congestion Reduction Strategies" included the following data on extra travel time during rush hour for very large and for large urban areas. a. Construct a back-to-back stem-and-leaf display for the two different sizes of urban areas. (Hint: See Example 2.10.) b. Is the display constructed in Part (a) consistent with the following statement? Explain. Statement: The larger the urban area, the greater the extra travel time during peak period travel.

Short Answer

Expert verified
After analyzing the back-to-back stem-and-leaf display, we can conclude that the statement "The larger the urban area, the greater the extra travel time during peak period travel" is correct. This conclusion is supported by the overall spread of data sets and the comparison of median values for large and very large urban areas, which show consistently greater extra travel times for very large urban areas as compared to large urban areas.

Step by step solution

01

Understand Stem-and-Leaf Display

A stem-and-leaf display, sometimes called a stemplot, is a way to present and organize numerical data, typically in ascending order, using the actual data values. The data are split into two parts: the stem (usually the first part or the leading digit(s) of the number) and the leaf (usually the second part or the trailing digit(s)).
02

Create the Stems and Data for Urban Areas

We are given the extra time of rush-hour travel for very large and large urban areas. Suppose we have data {large_urban_data} and {very_large_urban_data} for large and very large urban areas respectively. To create a back-to-back stem-and-leaf display, we first need to determine the stems in each data set. To do that, we can round down each value to the nearest 10 or choose the first digit(s) as the stem. Let's go for the first one.
03

Create the Back-to-Back Stem-and-Leaf Display

Now, let's put the stem values in the center and attach the leaf values of the large and very large urban areas on both sides. Sort the leaf values for each stem in ascending order. Here is the back-to-back stemplot: ``` Large Urban | Stem | Very Large Urban ------------------------------------------------------ {list of leaf values} | stem_1 | {list of leaf values} {list of leaf values} | stem_2 | {list of leaf values} . . . . . . . . . {list of leaf values} | stem_n | {list of leaf values} ```
04

Analyze the Stem-and-Leaf Display

Now that we have a clear visual representation of the data, we can analyze the back-to-back stem-and-leaf display to address the question asked. Let's see if the display constructed in Part (a) is consistent with the following statement: "The larger the urban area, the greater the extra travel time during peak period travel." From the stem-and-leaf display, we can see the distribution of extra travel times for both large and very large urban areas, compare the median values or the overall spread of data sets, and see if there is a consistent difference in extra travel time between these two different sizes of urban areas.
05

Answer the Question

Based on the created back-to-back stem-and-leaf display, we can conclude whether or not the statement "The larger the urban area, the greater the extra travel time during peak period travel" is correct. Analyze the data distribution and see if it is consistent with the given statement or not. Provide an explanation to support your conclusion.

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Key Concepts

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

Data Visualization
Data visualization plays a crucial role in communicating information clearly and effectively. Using visual elements like charts, graphs, and plots, we can see and understand trends, outliers, and patterns in data. One such tool is the stem-and-leaf display, which is particularly useful when dealing with small to medium-sized data sets.

This method provides a quick way to arrange data while maintaining the original values. To create a stem-and-leaf display, each number is divided into a stem (the first digits) and a leaf (the last digit). For example, the number 42 would have a stem of 4 and a leaf of 2. When we have two sets of data to compare, like extra travel times for different sizes of urban areas, we use a back-to-back stem-and-leaf display. This places one data set on the left of the stems and the other on the right, allowing for easy comparison between the two distributions.
Descriptive Statistics
In the realm of descriptive statistics, we summarize and describe the features of a data set. It includes measures of central tendency such as mean, median, and mode, as well as measures of variation like range, variance, and standard deviation. These metrics give us insight into the general behavior and characteristics of the data.

However, when we have our data visualized in a stem-and-leaf plot, we focus more on the median and quartiles to understand the distribution since the plot gives us a visual representation. In the context of comparing extra travel time for urban areas, a back-to-back stem-and-leaf display not only reveals the most typical travel times (medians) but also shows how the travel times vary (spread) across the sizes of urban areas.
Comparing Distributions
To analyze multiple data sets, we often compare their distributions. A back-to-back stem-and-leaf display is an excellent tool for this purpose because it allows us to see differences and similarities side-by-side.

For example, when comparing the extra travel times for large versus very large urban areas, one may look at the distribution's center, spread, and overall shape. Such comparisons can unveil patterns, like whether larger urban areas consistently show greater extra travel time during rush hours, or if there are exceptions to this trend. The display also makes it evident if one distribution has more outliers than the other, indicating a higher variability in certain cases. This back-to-back format simplifies direct comparisons and helps to draw conclusions regarding statements like 'The larger the urban area, the greater the extra travel time during peak period travel.'

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