91Ó°ÊÓ

The accompanying data set consists of observations on shear strength (lb) of ultrasonic spot welds made on a type of alclad sheet. Construct a relative frequency histogram based on ten equalwidth classes with boundaries \(4000,4200, \ldots .\) [The histogram will agree with the one in "Comparison of Properties of Joints Prepared by Ultrasonic Welding and Other Means" (J. Aircraft, 1983: 552-556).] Comment on its features. \(\begin{array}{lllllll}5434 & 4948 & 4521 & 4570 & 4990 & 5702 & 5241 \\ 5112 & 5015 & 4659 & 4806 & 4637 & 5670 & 4381 \\ 4820 & 5043 & 4886 & 4599 & 5288 & 5299 & 4848 \\ 5378 & 5260 & 5055 & 5828 & 5218 & 4859 & 4780 \\ 5027 & 5008 & 4609 & 4772 & 5133 & 5095 & 4618 \\ 4848 & 5089 & 5518 & 5333 & 5164 & 5342 & 5069 \\ 4755 & 4925 & 5001 & 4803 & 4951 & 5679 & 5256 \\ 5207 & 5621 & 4918 & 5138 & 4786 & 4500 & 5461 \\ 5049 & 4974 & 4592 & 4173 & 5296 & 4965 & 5170 \\ 4740 & 5173 & 4568 & 5653 & 5078 & 4900 & 4968 \\ 5248 & 5245 & 4723 & 5275 & 5419 & 5205 & 4452 \\ 5227 & 5555 & 5388 & 5498 & 4681 & 5076 & 4774 \\ 4931 & 4493 & 5309 & 5582 & 4308 & 4823 & 4417 \\ 5364 & 5640 & 5069 & 5188 & 5764 & 5273 & 5042 \\ 5189 & 4986 & & & & & \end{array}\)

Step by step solution

01

Organize Data in Ascending Order

First, we need to sort the provided data in ascending order. This will make it easier to group the data into the specified classes: 4173, 4308, 4381, 4417, 4452, 4493, 4500, 4521, 4568, 4570, 4592, 4599, 4609, 4618, 4637, 4659, 4681, 4723, 4740, 4755, 4772, 4774, 4780, 4786, 4803, 4806, 4820, 4823, 4848, 4848, 4859, 4886, 4900, 4918, 4925, 4931, 4948, 4951, 4965, 4968, 4974, 4986, 4990, 5001, 5008, 5015, 5027, 5042, 5043, 5049, 5055, 5069, 5069, 5076, 5078, 5089, 5095, 5112, 5133, 5138, 5164, 5170, 5173, 5188, 5189, 5205, 5207, 5218, 5227, 5241, 5245, 5248, 5256, 5260, 5273, 5275, 5288, 5296, 5299, 5309, 5333, 5342, 5364, 5378, 5388, 5419, 5434, 5461, 5498, 5518, 5555, 5582, 5621, 5640, 5653, 5670, 5679, 5702, 5764, 5828.

02

Determine Class Width

To construct classes with equal width, we use the given boundaries ranging from 4000 to over 5800, with steps of 200. Since 4000 isn't in the data, we start with the lowest data point 4173. Class boundaries: 1. 4000 - 4200 2. 4200 - 4400 3. 4400 - 4600 4. 4600 - 4800 5. 4800 - 5000 6. 5000 - 5200 7. 5200 - 5400 8. 5400 - 5600 9. 5600 - 5800 10. 5800 - 6000

03

Tally Data Into Classes

Now, we count how many data points fall into each class: 1. 4000 - 4200: 1 2. 4200 - 4400: 2 3. 4400 - 4600: 8 4. 4600 - 4800: 14 5. 4800 - 5000: 17 6. 5000 - 5200: 14 7. 5200 - 5400: 18 8. 5400 - 5600: 8 9. 5600 - 5800: 9 10. 5800 - 6000: 2

04

Calculate Relative Frequencies

Relative frequency for each class is calculated by dividing the frequency of each class by the total number of observations, which is 100. 1. 4000 - 4200: 0.01 2. 4200 - 4400: 0.02 3. 4400 - 4600: 0.08 4. 4600 - 4800: 0.14 5. 4800 - 5000: 0.17 6. 5000 - 5200: 0.14 7. 5200 - 5400: 0.18 8. 5400 - 5600: 0.08 9. 5600 - 5800: 0.09 10. 5800 - 6000: 0.02

05

Construct the Histogram

Using the relative frequencies calculated step by step, construct a relative frequency histogram. Each class boundary is represented along the x-axis and the relative frequency on the y-axis. The height of each bar corresponds to the relative frequency of that class.

06

Analyze the Histogram Features

The histogram is slightly right-skewed, indicating more data points at higher values. The greatest concentration of data is in the class 5200 - 5400 followed closely by 4800 - 5000 and 5000 - 5200, which is common in strength measurements, as medium-high values are more frequent. The shape suggests variability but with a higher occurrence of above-average values.

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Key Concepts

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

Data Sorting

Data sorting is the vital first step in constructing any histogram, especially a relative frequency histogram. By putting data in order, from the smallest to the largest value, we gain clarity and ease of use in later steps of analysis. For instance, in our exercise data related to shear strength of ultrasonic spot welds, sorting helps us see how the measurements spread out.
Sorting data doesn't change the data; it simply reorganizes it to make it more manageable. This sorted data can then be grouped into classes or intervals, which are required to form the relative frequency histogram.
This methodical approach ensures that all values are accounted for and can be appropriately categorized into respective class boundaries later on.

Class Boundaries

Class boundaries define the range of data that fits into each interval or class in a histogram. They are crucial for ensuring that we tally the data into meaningful groups.
In setting class boundaries, you are essentially setting intervals such as 4000 - 4200, 4200 - 4400, and so forth. These intervals ensure that each data point fits into one and only one class. It establishes an organized framework for subsequent steps, such as compiling the frequency and then calculating the relative frequency.
Deciding on appropriate class intervals depends on distribution and range of data. In our case, the given class boundaries have a width of 200, which helps in evenly spreading the data across ten classes from 4000 to 6000.

Relative Frequency Calculation

Once you have organized the data into classes, the next step is calculating relative frequency. This calculation expresses the number of observations in each class as a fraction of the total number of observations and is essential for creating a relative frequency histogram.
The formula for calculating the relative frequency of each class is: \[ \text{Relative Frequency} = \frac{\text{Frequency of the class}}{\text{Total number of observations}} \] This ratio helps in understanding the proportion of data points within each class compared to the whole.
For example, if a class holds 17 out of 100 total data points, the relative frequency would be 0.17. Calculating relative frequencies standardizes the data, allowing the histogram to show proportions rather than just counts.

Histogram Analysis

After constructing the histogram using data's relative frequencies, analyzing it is the final step. A histogram visually summarizes data distribution and provides insights into the dataset's underlying properties. Key aspects to examine include the shape, spread, and center of the distribution.
In the example dataset, the histogram reveals a slight right skew, indicating more higher shear strength values. This shape can help identify common trends and distributions in shear strength.
Histograms also make it easier to spot features such as mode (most common data range) or any anomalies, like gaps or outliers. It's a powerful visual tool that aids in understanding and communicating data characteristics effectively. The information gleaned from histogram analysis can inform decisions in practical applications, such as enhancing the quality of ultrasonic welding processes.