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Automated electron backscattered diffraction is now being used in the study of fracture phenomena. The following information on misorientation angle (degrees) was extracted from the article "Observations on the Faceted Initiation Site in the Dwell-Fatigue Tested Ti-6242 Alloy: Crystallographic Orientation and Size Effects (Metallurgical and Materials Trans., 2006: 1507-1518). $$ \begin{array}{lcccc} \text { Class: } & 0-<5 & 5-<10 & 10-<15 & 15-<20 \\ \text { Rel freq: } & .177 & .166 & .175 & .136 \\ \text { Class: } & 20-<30 & 30-<40 & 40-<60 & 60-<90 \\ \text { Rel freq: } & .194 & .078 & .044 & .030 \end{array} $$ a. Is it true that more than \(50 \%\) of the sampled angles are smaller than \(15^{\circ}\), as asserted in the paper? b. What proportion of the sampled angles are at least \(30^{\circ}\) ? c. Roughly what proportion of angles are between \(10^{\circ}\) and \(25^{\circ} ?\) d. Construct a histogram and comment on any interesting features.

Step by step solution

01

Calculate Proportion of Angles < 15°

To determine the proportion of sampled angles less than 15°, add the relative frequencies of the relevant classes. These classes are: 0-<5, 5-<10, and 10-<15. \[0.177 + 0.166 + 0.175 = 0.518\] Thus, 51.8% of the sampled angles are smaller than 15°.

02

Calculate Proportion of Angles ≥ 30°

Add the relative frequencies of the classes corresponding to angles at least 30°. These classes are: 30-<40, 40-<60, and 60-<90.\[0.078 + 0.044 + 0.030 = 0.152\]Thus, 15.2% of the sampled angles are at least 30°.

03

Calculate Proportion of Angles Between 10° and 25°

Examine the classes that overlap with the 10° to 25° range. The relevant classes are: 10-<15 (fully included) and a portion of 15-<20, and 20-<30. However, since the 20-<30 class also fully covers the 20-<25 span, we need the whole of the 15-<20 and a portion of the 20-<30.Calculate the proportion for 10-<15 and 15-<20:\[0.175 + 0.136 = 0.311\]This is the proportion of angles from 10° to 20°. Since 20-<30 is an overlapping category, from 20° to 25° take half of the 20-<30 interval:\[0.194/2 = 0.097\]Add to find the total from 10° to 25°:\[0.311 + 0.097 = 0.408\]Thus, approximately 40.8% of angles are between 10° and 25°.

04

Construct Histogram and Comment

To construct the histogram, plot the classes on the x-axis (0-<5, 5-<10, etc.) and their corresponding relative frequencies on the y-axis. The histogram would show two peaks: a larger peak between 10-<15 and 20-<30, and a smaller peak for 0-<5. The distribution appears somewhat bimodal, indicating variability in misorientation angles around these ranges.

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Key Concepts

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

Misorientation Angle Analysis

When studying materials like the Ti-6242 alloy used in various industrial applications, understanding the misorientation angle of crystals is crucial. The misorientation angle measures how much the orientation of one crystal lattice differs from another. In practical terms, it helps materials scientists understand how grains are oriented relative to each other, which influences properties like strength and fracture behavior.
An angle between 0° and a few degrees often indicates tightly packed and potentially stronger grain boundaries, while larger angles can lead to weaker points where fractures might initiate.
By analyzing misorientation angles, scientists can predict material behavior under stress and design more resilient materials.

Frequency Distribution

Frequency distribution is a method to summarize and present data in a structured way. It involves dividing data into classes or intervals and counting how often data points fall into each class. The classes in our exercise represent intervals of misorientation angles, such as 0-<5 degrees or 10-<15 degrees.
Each class is associated with a relative frequency, which is the proportion of observations within that class relative to the total observations. For instance, if the class 0-<5 degrees has a relative frequency of 0.177, this means 17.7% of misorientation angle measurements fall within that interval.
Frequency distribution helps in visualizing data trends and patterns, making it easier to interpret large datasets.

Histogram Construction

To build a histogram, we translate frequency distribution information into a graphical format. It's essentially a bar graph showing the frequency of data within certain intervals. Each bar represents a different class, where the height corresponds to the relative frequency of that class.
1. **Identify Classes**: For our data, classes like 0-<5 or 10-<15 degrees are identified.
2. **Plotting**: On the horizontal axis, draw the interval classes, and on the vertical axis, mark the relative frequency.
3. **Draw Bars**: Each bar's height represents its class’s relative frequency. Bars should touch each other to indicate continuous data.
The histogram for the misorientation angles data would show where most data falls, highlighting two peaks, making the pattern easy to see and analyze.

Bimodal Distribution

Bimodal distribution refers to a frequency distribution with two peaks. When data, like misorientation angles, are plotted in a histogram and show two distinct maxima, it's considered bimodal. This indicates that there are two prevalent groups or modes in the dataset.
In the context of misorientation angles, a bimodal distribution could suggest the presence of two dominant grain orientations or twin boundaries in the alloy.
This is important in materials science because it can affect the material properties like ductility or toughness. Knowing that a distribution is bimodal helps engineers and scientists infer the composition and interaction of different grain structures in the material.