Question

In a study of warp breakage during the weaving of fabric (Technometrics [1982]: 63), 100 pieces of yarn were tested. The number of cycles of strain to breakage was recorded for each yarn sample. The resulting data are given in the following table: $$ \begin{array}{rrrrrrrrrr} 86 & 146 & 251 & 653 & 98 & 249 & 400 & 292 & 131 & 176 \\ 76 & 264 & 15 & 364 & 195 & 262 & 88 & 264 & 42 & 321 \\ 180 & 198 & 38 & 20 & 61 & 121 & 282 & 180 & 325 & 250 \\ 196 & 90 & 229 & 166 & 38 & 337 & 341 & 40 & 40 & 135 \\ 597 & 246 & 211 & 180 & 93 & 571 & 124 & 279 & 81 & 186 \\ 497 & 182 & 423 & 185 & 338 & 290 & 398 & 71 & 246 & 185 \\ 188 & 568 & 55 & 244 & 20 & 284 & 93 & 396 & 203 & 829 \\ 239 & 236 & 277 & 143 & 198 & 264 & 105 & 203 & 124 & 137 \\ 135 & 169 & 157 & 224 & 65 & 315 & 229 & 55 & 286 & 350 \\ 193 & 175 & 220 & 149 & 151 & 353 & 400 & 61 & 194 & 188 \end{array} $$ a. Construct a frequency distribution using the class intervals 0 to \(<100,100\) to \(<200\), and so on. b. Draw the histogram corresponding to the frequency distribution in Part (a). How would you describe the shape of this histogram? c. Find a transformation for these data that results in a more symmetric histogram than what you obtained in Part (b).

Short answer

After undergoing these steps, the histogram displays the frequency of cycles of strain to breakage in each class of values. After we find the most symmetric histogram through transformation, we can say that the shape of the histogram has been significantly improved.

Step-by-step solution

Construct Frequency Distribution

First, let's construct a frequency distribution using the class intervals given. The class intervals are 0 to 100, 100 to 200 and so on till 900. Count how many numbers fall into each of these intervals. This can be done manually, or using a computational tool.

Create a Histogram

Based on the frequency distribution created in Step 1, plot a histogram. The x-axis of the histogram represents the class intervals (or 'bins') and the heights of the bars represent the frequency (number of yarn samples) for each class.

Analyze Histogram Shape

Analyze the shape of the histogram. It might be skewed left or right, or could be multimodal. Describing this shape is subjective and depends on the resulting histogram from your data.

Find a Transformation for the Data

In order to get a more symmetric histogram, the given data might need to be transformed. Common transformations include taking the square root, the logarithm, or the reciprocal of each data point. Try these transformations and create histograms for them. The most symmetric histogram would be the desired transformation.

Analyze Transformed Histogram

Once the transformation is applied to the data, analyze the shape of the resultant histogram again. The shape should be more symmetric compared to the previous one.

Key concepts

Histogram

A histogram is a graphical representation that organizes a group of data points into user-specified ranges, known as bins or classes. Imagine it as a bar chart that shows the frequency of occurrences within these intervals. The construction of a histogram involves dividing the entire range of values into a series of intervals and then counting how many values fall into each interval. The frequency distribution, as featured in the exercise given, is at the core of building a histogram.

In the exercise, we were asked to create a histogram with class intervals of 100 (0 to <100, 100 to <200, and so on). By plotting the frequency of yarn breakage cycles within these intervals, one can easily visualize the distribution of breakage occurrences. This visual tool is pivotal in statistical data analysis, as it helps to reveal the underlying frequency distribution of a data set at a glance. After plotting the histogram based on the yarn data, we might discover patterns such as skewness or peaks, which are indicative of the data's spread and central tendency.

Data Transformation

Data transformation is a statistical technique used to convert data from one format or structure to another. The primary goal is often to achieve a more symmetrical or normal distribution, which can simplify analysis and improve the validity of statistical tests. Common transformations include taking logarithms, square roots, or reciprocals of the data points.

In the exercise, we seek a transformation that yields a histogram that is more symmetric than the original. This may be necessary if the initial histogram exhibits substantial skewness, indicating that the data is not normally distributed. For example, if a dataset is right-skewed, applying a logarithmic transformation can help to 'pull in' large values and moderate the skew. Similarly, if data contains non-negative values, a square root transformation can stabilize variance and make the dataset more amenable to analysis. In the exercise improvement advice, applying such transformations helps in achieving a dataset that, when visualized in a histogram, appears more balanced around its central value.

Statistical Data Analysis

Statistical data analysis encompasses the techniques and processes used to understand the behavior of data and draw conclusions. In the context of the original exercise, once a histogram is constructed, it's essential to analyze its shape. The shape can be symmetric, skewed to the right (positively skewed), skewed to the left (negatively skewed), or even bimodal (having two peaks). These characteristics give us insights into the distribution of data: a symmetrical histogram suggests a more 'normal' distribution, while skewness might prompt further investigation into outliers or data transformation.

The ability to choose the appropriate transformation and understand its effects on the data is an invaluable part of statistical data analysis. Demonstrating how the data behaves through a symmetric histogram post-transformation, as shown in the final step of the solution, helps provide clarity on the distribution's characteristics. It's essential to interpret the transformed data carefully, as the transformation must also be considered when making any inferences or conclusions based on the analysis.