91Ó°ÊÓ

In a study of warp breakage during the weaving of fabric (Technometrics, 1982: 63), 100 specimens of yarn were tested. The number of cycles of strain to breakage was determined for each yarn specimen, resulting in the following data: \(\begin{array}{rrrrrrrrrr}86 & 146 & 251 & 653 & 98 & 249 & 400 & 292 & 131 & 169 \\ 175 & 176 & 76 & 264 & 15 & 364 & 195 & 262 & 88 & 264 \\ 157 & 220 & 42 & 321 & 180 & 198 & 38 & 20 & 61 & 121 \\ 282 & 224 & 149 & 180 & 325 & 250 & 196 & 90 & 229 & 166 \\ 38 & 337 & 65 & 151 & 341 & 40 & 40 & 135 & 597 & 246 \\ 211 & 180 & 93 & 315 & 353 & 571 & 124 & 279 & 81 & 186 \\ 497 & 182 & 423 & 185 & 229 & 400 & 338 & 290 & 398 & 71 \\ 246 & 185 & 188 & 568 & 55 & 55 & 61 & 244 & 20 & 284 \\ 393 & 396 & 203 & 829 & 239 & 236 & 286 & 194 & 277 & 143 \\ 198 & 264 & 105 & 203 & 124 & 137 & 135 & 350 & 193 & 188\end{array}\) a. Construct a relative frequency histogram based on the class intervals \(0-100,100-200, \ldots\), and comment on features of the distribution. b. Construct a histogram based on the following class intervals: \(0-50,50-100,100-150\), \(150-200, \quad 200-300, \quad 300-400, \quad 400-500\), \(500-600,600-900 .\) c. If weaving specifications require a breaking strength of at least 100 cycles, what proportion of the yam specimens in this sample would be considered satisfactory?

Step by step solution

01

Organize Data into Intervals for Part (a)

To construct a relative frequency histogram, first divide the data into specified class intervals: \(0-100, 100-200, 200-300,\) etc. Count the number of data points that fall into each interval. For example, count the number of values between 0 and 100, 100 and 200, and so on.

02

Calculate Relative Frequencies for Part (a)

For each interval calculated in Step 1, determine the relative frequency by dividing the number of data points in the interval by the total number of data points (100). This will give you a fraction or percentage representing the portion of the total data within each interval.

03

Construct Histogram for Part (a)

Plot the relative frequencies calculated in Step 2 on the y-axis against the intervals on the x-axis. Each bar in the histogram corresponds to a different interval, with the height representing the relative frequency of that interval.

04

Comment on Distribution for Part (a)

Observe the shape of the histogram from Step 3. Comment on features such as whether the distribution appears symmetric, skewed, or if there are any noticeable peaks or clusters.

05

Organize Data into Intervals for Part (b)

For the histogram in Part (b), use the class intervals: \(0-50, 50-100, 100-150,\) etc. Again, count how many data points fall into each of these new intervals.

06

Construct Histogram for Part (b)

Based on the counts from Step 5, plot another histogram. This histogram will use the new intervals with the counts plotted similarly to Step 3.

07

Evaluate Breaking Specification for Part (c)

To determine what proportion of the yarn specimens meet the specification of at least 100 cycles, count the number of data points that are greater than or equal to 100. Divide this count by the total number of specimens (100) to find the proportion that meets the requirement.

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Key Concepts

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

Relative Frequency

Relative frequency is a way to understand how often something happens relative to the total number of observations. It's like finding out what part of a whole your specific interest represents. In the case of building a relative frequency histogram from yarn breakage data, it involves a few steps:

• First, the data must be organized into intervals; these are the 'class intervals' like \(0-100, 100-200\), etc.
• Next, count how many data points fall into each class interval.
• Then, for each interval, divide the number of data points by the total number of samples (in this case, 100) to find the relative frequency.

This gives a fraction for each interval representing its share of the total. In simpler terms, if 10 yarn samples fall into the 0-100 cycle interval, and there are 100 samples total, the relative frequency is \(10/100 = 0.1\) or 10%.

This step forms the basis of constructing the histogram, which allows you to visually understand the distribution of breaking cycles in a dataset, with each bar height reflecting relative frequency.

Data Distribution

Data distribution shows us how data points are spread across different intervals or categories. Looking at a histogram can tell us a lot about this spread. When you build a histogram for the yarn breakage data, the following things can be observed:

• Histograms show peaks which indicate intervals where data points are more concentrated.
• They can also show if the data is symmetric or skewed in either direction. For instance, if most data fall on the lower intervals, it might indicate left skewness.
• Clusters or gaps can be detected, indicating regions of frequent or rare data points.

Understanding data distribution is key in analytics as it helps identify patterns or anomalies in data. In the context of yarn breakage, this distribution could highlight both expected and unexpected behaviors of the yarn strength under stress.

By using different sets of intervals, like in original parts (a) versus (b), one can gain a deeper insight into specific areas of interest, like more focus on lower breaking strengths.

Breaking Strength Analysis

Breaking strength analysis refers to understanding how much stress a yarn can take before breaking. It's a crucial part of quality control in textiles. In the yarn breakage study, it's essential to focus on a critical question: which yarn samples meet a standard of breaking strength?

• The specific requirement here is that a satisfactory yarn must withstand at least 100 cycles. So, meaning we want to know how many samples have breaking cycles of 100 or more.
• To determine this, count all data points that are \(\geq 100\).
• Finally, divide by the total number of samples to get a proportion, indicating how much of the yarn is satisfactory according to this criterion.

This analysis is not only about meeting standards but also about enhancing quality control measures. Knowing what proportion of products pass or fail gives manufacturers actionable insights into production processes and areas that require improvements. It might lead to further investigations or adjustments in the weave or yarn composition to ensure better performance.