91Ó°ÊÓ

Allowable mechanical properties for structural design of metallic aerospace vehicles requires an approved method for statistically analyzing empirical test data. The article "Establishing Mechanical Property Allowables for Metals" (J. of Testing and Evahuation, 1998: 293-299) used the accompanying data on tensile ultimate strength (ksi) as a basis for addressing the difficulties in developing such a method. \(\begin{array}{lllllllll}122.2 & 124.2 & 124.3 & 125.6 & 126.3 & 126.5 & 126.5 & 127.2 & 127.3 \\ 127.5 & 127.9 & 128.6 & 128.8 & 129.0 & 129.2 & 129.4 & 129.6 & 130.2 \\ 130.4 & 130.8 & 131.3 & 131.4 & 131.4 & 131.5 & 131.6 & 131.6 & 131.8 \\ 131.8 & 132.3 & 132.4 & 132.4 & 132.5 & 132.5 & 132.5 & 132.5 & 132.6\end{array}\) \(\begin{array}{lllllllll}132.7 & 132.9 & 133.0 & 133.1 & 133.1 & 133.1 & 133.1 & 133.2 & 133.2 \\ 133.2 & 133.3 & 133.3 & 133.5 & 133.5 & 133.5 & 133.8 & 133.9 & 134.0 \\ 134.0 & 134.0 & 134.0 & 134.1 & 134.2 & 134.3 & 134.4 & 134.4 & 134.6 \\ 134.7 & 134.7 & 134.7 & 134.8 & 134.8 & 134.8 & 134.9 & 134.9 & 135.2 \\ 135.2 & 135.2 & 135.3 & 135.3 & 135.4 & 135.5 & 135.5 & 135.6 & 135.6 \\ 135.7 & 135.8 & 135.8 & 135.8 & 135.8 & 135.8 & 135.9 & 135.9 & 135.9 \\ 135.9 & 136.0 & 136.0 & 136.1 & 136.2 & 136.2 & 136.3 & 136.4 & 136.4 \\ 136.6 & 136.8 & 136.9 & 136.9 & 137.0 & 137.1 & 137.2 & 137.6 & 137.6 \\ 137.8 & 137.8 & 137.8 & 137.9 & 137.9 & 138.2 & 138.2 & 138.3 & 138.3 \\ 138.4 & 138.4 & 138.4 & 138.5 & 138.5 & 138.6 & 138.7 & 138.7 & 139.0 \\ 139.1 & 139.5 & 139.6 & 139.8 & 139.8 & 140.0 & 140.0 & 140.7 & 140.7 \\ 140.9 & 140.9 & 141.2 & 141.4 & 141.5 & 141.6 & 142.9 & 143.4 & 143.5 \\ 143.6 & 143.8 & 143.8 & 143.9 & 144.1 & 144.5 & 144.5 & 147.7 & 147.7\end{array}\) a. Construct a stem-and-leaf display of the data by first deleting (truncating) the tenths digit and then repeating each stem value five times (once for leaves 1 and 2 , a second time for leaves 3 and 4 , etc.). Why is it relatively easy to identify a representative strength value? b. Construct a histogram using equal-width classes with the first class having a lower limit of 122 and an upper limit of 124 . Then comment on any interesting features of the histogram.

Step by step solution

01

Organize Tensile Strength Data

First, we prepare the data without the tenths place, effectively truncating the tensile strength values to their integer stems. The modified data will be used for the stem-and-leaf display in Part A.

02

Create a Stem-and-Leaf Display

To construct the stem-and-leaf display, repeat the integer stem five times. Each stem corresponds to two leaf values: 1-2, 3-4, etc. For example, a stem "125" will appear multiple times: once for leaves 0-2, once for 3-4, and so forth until 9-9. This elongates the stem effectively to display the distribution of the tensile strengths.

03

Design Stem-and-Leaf Display

With the truncated values, create a stem-and-leaf plot with stems repeated to cover all leaf groupings (e.g., 1-2, 3-4, etc.). Organize the leaves appropriately under each stem repetition. Representative strength is easy to spot as clusters in the middle section around frequent values.

04

Group Data for Histogram

For Part B, prepare data for a histogram. Bin the data with class widths that range from 122 to 124, 124 to 126, etc. Count how many data points fall within each interval.

05

Create Histogram

Plot a histogram with each bin interval on the horizontal axis and frequency of tensile strength values within each interval on the vertical axis. Initiate the first class from 122 to 124 and continue with equal-width bins upwards until all data are assigned.

06

Analyze and Comment on Histogram

Examine the histogram for patterns. Note any central tendency, variation, or anomalies. A concentration of data implies a clear central tendency. Analyze interesting segments like skewness or heavy tails and comment on what this suggests about tensile strength characteristics.

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Key Concepts

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

Stem-and-Leaf Display

A stem-and-leaf display is a simple and informative method for displaying numerical data. It lists data points by splitting each value into a "stem" and a "leaf." The stem represents the leading digits, while the leaf shows the trailing part. This method provides a visual overview of data distribution while retaining the original data values.
In the exercise at hand, the process starts with truncating the tensile strength values, removing the tenths digit, to focus on integer values. This reduced form helps create an effective stem-and-leaf display:

• The stem portion uses the main digits (e.g., 127 becomes 12, with 7 acting as the leaf).
• Each stem is repeated five times in this particular exercise to ensure each pair of leaf values (1-2, 3-4, etc.) can be clearly represented.

By organizing the data in this way, patterns and trends become more apparent. For instance, if many leaves gather around stems in the middle of the dataset, this indicates a central area of value concentration. This stem-and-leaf format is especially useful for small to medium-sized data sets. It visually highlights clusters and outliers without losing the abstract view of the dataset.

Histogram Construction

Histograms play a crucial role in summarizing group data for statistical analysis. They provide a way to visualize the distribution of a data set, showing the frequency of data points within defined bins or intervals.
In constructing a histogram from this exercise, the following steps are taken:

• First, determine the class intervals, such as 122 to 124, 124 to 126, etc. These intervals need to be of equal width and cover the entire dataset.
• Count how many data points fall within each class interval to prepare the necessary data for plotting.

Once prepared, the histogram is plotted with class intervals on the x-axis and frequencies on the y-axis. Each bar height correlates with the frequency of data points in its class interval.
Analyzing the histogram lets us spot patterns like:

• Clusters or concentrations of data, indicating a central tendency.
• Variations or spread revealing the dispersion of data points.
• Skewness or tails suggesting asymmetry in data distribution.

These insights help engineers and statisticians understand underlying characteristics, such as typical tensile strength values.

Mechanical Property Analysis

Analyzing mechanical properties is critical in engineering, particularly for applications like aerospace design where structural integrity is paramount. Understanding the mechanical properties of metals, such as tensile strength, helps design materials that can withstand specific loads without failure.
In the context of this exercise, tensile strength results are obtained and statistically analyzed to develop or verify existing methods for determining property allowables. Key aspects of this analysis include:

• Ensuring data accuracy and representative sampling to reflect real-world material behavior.
• Employing statistical methods, like those illustrated, to grasp overall properties and variations.

Mechanical property analysis provides engineers with insights into how a material will react under stress. It aligns with safety standards and regulations, ensuring the materials used in aerospace applications meet stringent demands. Correctly interpreting and applying this data informs decisions on material selection, design optimizations, and performance approximations.