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 histogram. b. Construct a histogram based on the following class intervals: \(0-<50, \quad 50-<100, \quad 100-<150, \quad 150-<200\), \(200-<300,300-<400,400-<500,500-<600\), and \(600-<900 .\) c. If weaving specifications require a breaking strength of at least 100 cycles, what proportion of the yarn specimens in this sample would be considered satisfactory?

Step by step solution

01

Organize the Data

The data consists of 100 values representing the number of cycles to breakage for yarn specimens. To create histograms, we need to group these data points into specified class intervals that will simplify the data visualization.

02

Count Frequencies for Each Class Interval (Part a)

To construct the histogram for part (a) with intervals such as \(0-<100\), \(100-<200\), ..., continue counting frequencies: - \([0,100)\): Count the number of values less than 100.- \([100,200)\): Count the number of values between 100 and less than 200.Continue this process for \([200,300)\), \([300,400)\), and so on.

03

Calculate Relative Frequencies (Part a)

Divide the frequency of each class interval by the total number of observations (100) to get the relative frequency. For example, if the \([0,100)\) interval contains 15 observations, the relative frequency would be \(\frac{15}{100} = 0.15\). Repeat this for all intervals.

04

Construct the Relative Frequency Histogram (Part a)

On the x-axis, use the class intervals as labels. On the y-axis, plot the relative frequencies for each class interval. Create bars for each interval whose height corresponds to its relative frequency. Comment on the shape and distribution of the data, noting features like skewness or bimodality.

05

Count Frequencies for New Intervals (Part b)

Follow the same procedure as Step 2 but with these new intervals: \([0,50)\), \([50,100)\), \([100,150)\), ..., \([600,900)\). Count the number of data points in each interval.

06

Construct the Frequency Histogram (Part b)

Using the frequencies calculated from Step 5, create a histogram with class intervals on the x-axis and frequency on the y-axis. Draw bars for each class interval.

07

Determine Satisfactory Proportion (Part c)

Identify the number of data points with breaking strength of at least 100 cycles. Count values from the interval \([100,200, \dots, 900)\). Add these counts to reflect the 'satisfactory' specimens. Divide the count by the total number of specimens (100) to find the proportion satisfying the condition.

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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 an essential concept in statistics used to understand how often an event occurs in relation to the total number of observations. It provides insight into the proportion of data points falling within a specific interval and is especially useful when dealing with large datasets.
To calculate the relative frequency, follow this procedure:

• Count the number of data points in each class interval.
• Divide this count by the total number of observations to get a proportion.

For instance, if there are 15 yarn specimens in the interval \(0-<100\), the relative frequency is calculated as \(\frac{15}{100} = 0.15\). The result is a value ranging from 0 to 1 that indicates the percentage of data points within that class interval. This metric helps visualize data distribution, making it easier to identify patterns and trends.

Class Intervals

Class intervals, also known as bins or groups, are range segments into which data is divided for analysis and visualization. They help in condensing vast data points into simpler categories that are easier to analyze, especially when creating histograms.
In practice, class intervals are selected based on data range and desired granularity. For example, in our exercise, the intervals \(0-<100\), \(100-<200\), etc., divide the data into usable segments. Often, intervals should be equal in size, but varying interval sizes can be used to highlight specific ranges.
When selecting class intervals, consider:

• The range of data values.
• The number of observations.
• The level of detail required in the analysis.

This approach helps create clear and informative visualizations that accurately reflect the underlying data distribution.

Data Visualization

Data visualization transforms complex data sets into more straightforward graphical representations, making it easier to interpret patterns, trends, and outliers. Histograms are a popular tool in data visualization that use bars to showcase data distribution across class intervals.
When constructing a histogram, each bar represents the frequency or relative frequency of the data within a specific interval on the horizontal axis, while the vertical axis represents frequency count or relative proportion. This allows for a quick assessment of data features, such as:

• Skewness - asymmetry in data distribution.
• Bi-modality - presence of two peaks in the dataset.
• Uniformity - even distribution across intervals.

By visualizing data as a histogram, you can quickly identify the spread of data over the ranges and assess how the data aligns with expected or desired results.

Proportion Calculation

Proportion calculation involves determining what fraction of the dataset meets certain criteria, providing a clear understanding of data compliance or satisfaction regarding specific conditions.
In the given problem, calculating the proportion of yarn specimens that meet the minimum breaking strength of 100 cycles involves the following steps:

• Count the number of specimens whose breaking strength is 100 cycles or more.
• Divide this count by the total number of specimens (in this case, 100).

For example, if 85 specimens break at or above 100 cycles, the proportion would be \(\frac{85}{100} = 0.85\). This value indicates that 85% of specimens are considered satisfactory. Proportions provide a simple yet powerful measure to evaluate data against standards or thresholds.