91Ó°ÊÓ

In the study of textiles and fabrics, the strength of a fabric is a very important consideration. Suppose that a significant number of swatches of a certain fabric are subjected to different "loads" or forces applied to the fabric. The data from such an experiment might look as follows: $$ \begin{aligned} &\text { Hypothetical Data on Fabric Strength }\\\ &\begin{array}{lccccccc} \hline \begin{array}{l} \text { Load } \\ \text { (lb/sq in.) } \end{array} & \mathbf{5} & \mathbf{1 5} & \mathbf{3 5} & \mathbf{5 0} & \mathbf{7 0} & \mathbf{8 0} & \mathbf{9 0} \\ \hline \begin{array}{l} \text { Proportion } \\ \text { failing } \end{array} & 0.02 & 0.04 & 0.20 & 0.23 & 0.32 & 0.34 & 0.43 \\ \hline \end{array} \end{aligned} $$ a. Make a scatterplot of the proportion failing versus the load on the fabric. b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the loads and fit the line \(y^{\prime}=a+b\) (Load). What is the significance of a positive slope for this line? c. What proportion of the time would you estimate this fabric would fail if a load of \(60 \mathrm{lb} / \mathrm{sq}\) in. were applied? d. In order to avoid a "wardrobe malfunction," one would like to use fabric that has less than a \(5 \%\) chance of failing. Suppose that this fabric is our choice for a new shirt. To have less than a \(5 \%\) chance of failing, what would you estimate to be the maximum "safe" load in \(\mathrm{lb} / \mathrm{sq}\) in.?

Step by step solution

01

Creating a Scatterplot

In this step, using a plotter, each load value is plotted in the X-axis against the proportion failing in the Y-axis. This gives the initial idea of the relationship between load and the proportion failing.

02

Logarithmic Transformation

The goal now is to transform the data to make it possible to apply linear regression. This is done by calculating for each proportion p, \(y' = ln(\frac{p}{1-p})\). Thus, each proportion failing gets converted to this new format, yielding another column of data.

03

Line of Best Fit

After the transformation all (Load,\(y'\)) pairs are plotted and the line of best fit is calculated. This line of best fit has the form \(y^' = a + b * \text{Load}\) where a and b are the slope and y-intercept obtained from the line of best fit.

04

Significance of Positive Slope

A positive slope in this case shows that the probability of fabric failure increases as load increases. It signifies a direct relationship between the load and the chance of fabric failure.

05

Predicting Failure Rate for 60 lb/sq in.

The equation of the line of best fit is used to predict this. Plugging in 60 for the Load in the equation \(y^' = a + b * \text{Load}\) gives the estimated failure rate.

06

Estimating Maximum Safe Load

To make this estimation, first convert 5% failing proportion to the transformed form using \(y' = ln(\frac{p}{1-p})\). This gives the \(y'\) for 5% failing. Then it's plugged into the equation \(y^' = a + b * \text{Load}\) to solve for the Load, which is the maximum safe load.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Scatterplot

A scatterplot is a type of data visualization that uses dots to represent the values obtained for two different variables – one plotted along the x-axis and the other plotted along the y-axis. This kind of plot is essential for exploring the relationship between two quantitative variables. In our case, the scatterplot would display each different load applied to fabric on the x-axis and the corresponding proportion of fabric that fails under that load on the y-axis.

Looking at a scatterplot, one can easily spot patterns, trends, and potential outliers in the data. In our exercise, the scatterplot not only reveals any linear or non-linear relationships between load and failure rate but also assists in assessing the strength and direction of the correlation between these two variables. Making a scatterplot is the starting point that sets the stage for further statistical analysis, such as fitting a trendline that could predict future outcomes.

Logarithmic Transformation

When we deal with a non-linear relationship between two variables, it's often useful to perform a logarithmic transformation, especially if we plan to fit a linear model to our data for predictive analysis. The logarithmic transformation is a powerful mathematical tool that can convert a non-linear relationship into a linear one by applying a log function to the data.

In the context of our fabric strength analysis, the transformation applied is \(y' = \ln\bigg(\frac{p}{1-p}\bigg)\), where \(p\) is the proportion of fabric failing. This specific transformation, known as the logit transformation, is commonly used in logistic regression and helps to linearize data where the dependent variable is a probability or a proportion, as is the case here. This linearization enables us to use linear regression techniques to find the best-fit line and make meaningful predictions.

Line of Best Fit

In statistical analysis, the line of best fit, also known as a trendline, is a straight line drawn through the center of a group of data points plotted on a scatterplot. This line summarizes the data, showing the relationship between the two variables. The equation of the line of best fit is \(y' = a + b \times \text{Load}\), where \(a\) is the y-intercept and \(b\) is the slope of the line. For our exercise, the line of best fit represents the adjusted equation after conducting the logarithmic transformation.

The slope of the line (\(b\)) gives us the rate of change between the load and the transformed failure rates. A positive slope indicates that as the load on the fabric increases, the likelihood of fabric failure also increases. Finding the best-fit line is crucial in predicting the failure rate of fabric for loads that weren't explicitly tested in the experiment.

Probability of Fabric Failure

Understanding the probability of fabric failure is critical to ensuring the safety and reliability of textile products. The probability can give us the expected rate at which a certain fabric might fail under different loads. Once we have the transformed data and the line of best fit, we can use this model to make predictions and calculate the likelihood of failure for any given load. For instance, we could estimate what proportion of fabric is likely to fail at \(60 \text{lb/sq in.}\) by plugging this load value into our best-fit line equation.

Furthermore, determining a 'safe' load – one that would result in a probability of failure below a certain threshold, like 5% – is essential for applications where fabric strength is paramount. The probability model derived from our analysis provides manufacturers and designers with the necessary information to set these thresholds and ensure that the fabric used in their products will perform safely under expected loads.