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Chapter 14: Problem 7

The article "Pulp Brightness Reversion: Influence of Residual Lignin on the Brightness Reversion of Bleached Sulfite and Kraft Pulps" (TAPPI \([1964]: 653-662)\) proposed a quadratic regression model to describe the relationship between \(x=\) degree of delignification during the processing of wood pulp for paper and \(y=\) total chlorine content. Suppose that the actual model is $$ y=220+75 x-4 x^{2}+e $$ a. Graph the regression function \(220+75 x-4 x^{2}\) over \(x\) values between 2 and \(12 .\) (Substitute \(x=2,4,6,8,10\), and 12 to find points on the graph, and connect them with a smooth curve.) b. Would mean chlorine content be higher for a degree of delignification value of 8 or \(10 ?\) c. What is the change in mean chlorine content when the degree of delignification increases from 8 to \(9 ?\) From 9 to \(10 ?\)

Short Answer

Expert verified
The graph of the regression function will be a downward-opening parabola because of the negative coefficient of the \(x^2\) term. The mean chlorine content decreases as the degree of delignification increases from 8 to 10. The change in mean chlorine content as the degree of delignification increases from 8 to 9 and from 9 to 10 can be calculated by subtracting the corresponding mean chlorine content for these values.

Step by step solution

01

Graphing the regression function

The first step is to graph the regression function \(220+75 x-4 x^{2}\). To do this, substitute \(x=2,4,6,8,10\), and 12 into the function and calculate the corresponding \(y\) values. Plot the \(x, y\) coordinates on a graph and connect them with a smooth curve to represent the regression function.
02

Comparing mean chlorine content

The next step is to compare the mean chlorine content for a degree of delignification value of 8 and 10. Substitute \(x=8\) and \(x=10\) into the regression model to find the mean chlorine content for each value and compare them.
03

Determining the change in mean chlorine content

To find the change in mean chlorine content when the degree of delignification increases from 8 to 9, and from 9 to 10, substitute \(x=8\) and \(x=9\) into the regression model to find the mean chlorine content for each value. The difference in these two quantities will give the change in mean chlorine content when the degree of delignification increases from 8 to 9. Repeat the process for \(x=9\) and \(x=10\) to find the change in mean chlorine content when the degree of delignification increases from 9 to 10.

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Key Concepts

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

Degree of Delignification
When it comes to paper production, the term degree of delignification refers to the process of removing lignin from wood pulp. Lignin is a complex organic polymer found in the cell walls of plants, imparting rigidity and preventing the collapse of the cell structure. However, lignin can cause paper to become yellow or brittle over time, making its removal a crucial step in the production of high-quality paper.

Delignification is typically achieved through chemical processes, often involving oxidative agents. The degree to which this process is carried out can affect various properties of the final paper product, including its brightness and strength. Understanding the degree of delignification is essential for improving the paper's quality and for predicting how it will react to bleaching agents like chlorine, which is often addressed in statistical models.
Chlorine Content in Wood Pulp
The chlorine content in wood pulp is an important variable in the paper industry. Chlorine and its compounds are used in the bleaching process to improve the whiteness and brightness of paper. However, the residual chlorine in the wood pulp can lead to the formation of harmful environmental pollutants such as dioxins.

Therefore, managing the chlorine content during the bleaching process is not only crucial for the quality of the paper but also for environmental sustainability. The relationship between the degree of delignification and the chlorine content is a key focus of study, as it allows manufacturers to fine-tune the paper-making process to minimize environmental impact while achieving desired product quality.
Statistical Analysis
Statistical analysis is a foundational tool across many fields, including the study of chemical processes in paper production. It involves collecting data, analyzing it, and drawing conclusions about the data set or the phenomena it reflects. In the context of paper production, statistical analyses can be used to establish relationships between different variables, such as the degree of delignification and chlorine content.

Quadratic regression models are a type of statistical analysis that can describe complex, non-linear relationships between variables. They can be critical for predicting outcomes and making informed decisions in manufacturing processes. High-quality statistical analysis allows for the improvement of product quality and optimization of production efficiency.
Graphing Regression Functions
Graphing regression functions visualizes the relationship between variables in a statistical model. For quadratic regression, the function typically involves a squared term, which produces a parabolic curve on the graph.

To graph a quadratic regression function, as in the exercise, one must first calculate the y-values for given x-values and plot these points on a graph. The resulting shape provides insights into how changes in one variable affect another. For example, it can show how changes in the degree of delignification affect the chlorine content in wood pulp. Such graphs are crucial in both analytical studies and in application within the production environment, facilitating a better understanding of the process dynamics and potential optimization.

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Most popular questions from this chapter

For the multiple regression model in Exercise \(14.4\), the value of \(R^{2}\) was \(.06\) and the adjusted \(R^{2}\) was \(.06 .\) The model was based on a data set with 1136 observations. Perform a model utility test for this regression.

The following statement appeared in the article "Dimensions of Adjustment Among College Women" (Journal of College Student Development \([1998]: 364):\) Regression analyses indicated that academic adjustment and race made independent contributions to academic achievement, as measured by current GPA. Suppose $$ \begin{aligned} y &=\text { current GPA } \\ x_{1} &=\text { academic adjustment score } \\ x_{2} &=\text { race (with white }=0 \text { , other }=1) \end{aligned} $$ What multiple regression model is suggested by the statement? Did you include an interaction term in the model? Why or why not?

The article "Impacts of On-Campus and Off-Campus Work on First-Year Cognitive Outcomes" (Journal of College Student Development \([1994]: 364-370\) ) reported on a study in which \(y=\) spring math comprehension score was regressed against \(x_{1}=\) previous fall test score, \(x_{2}=\) previous fall academic motivation, \(x_{3}=\) age, \(x_{4}=\) number of credit hours, \(x_{5}=\) residence \((1\) if on campus, 0 otherwise), \(x_{6}=\) hours worked on campus, and \(x_{7}=\) hours worked off campus. The sample size was \(n=210\), and \(R^{2}=.543\). Test to see whether there is a useful linear relationship between \(y\) and at least one of the predictors.

This exercise requires the use of a computer package. The authors of the article "Absolute Versus per Unit Body Length Speed of Prey as an Estimator of Vulnerability to Predation" (Animal Behaviour [1999]: \(347-\) 352) found that the speed of a prey (twips/s) and the length of a prey (twips \(\times 100\) ) are good predictors of the time (s) required to catch the prey. (A twip is a measure of distance used by programmers.) Data were collected in an experiment where subjects were asked to "catch" an animal of prey moving across his or her computer screen by clicking on it with the mouse. The investigators varied the length of the prey and the speed with which the prey moved across the screen. The following data are consistent with summary values and a graph given in the article. Each value represents the average catch time over all subjects. The order of the various speed-length combinations was randomized for each subject. $$ \begin{array}{ccc} \begin{array}{c} \text { Prey } \\ \text { Length } \end{array} & \begin{array}{l} \text { Prey } \\ \text { Speed } \end{array} & \begin{array}{l} \text { Catch } \\ \text { Time } \end{array} \\ \hline 7 & 20 & 1.10 \\ 6 & 20 & 1.20 \\ 5 & 20 & 1.23 \\ 4 & 20 & 1.40 \\ 3 & 20 & 1.50 \\ 3 & 40 & 1.40 \\ 4 & 40 & 1.36 \\ 6 & 40 & 1.30 \\ 7 & 40 & 1.28 \\ 7 & 80 & 1.40 \\ 6 & 60 & 1.38 \\ 5 & 80 & 1.40 \\ 7 & 100 & 1.43 \\ 6 & 100 & 1.43 \\ 7 & 120 & 1.70 \\ 5 & 80 & 1.50 \\ 3 & 80 & 1.40 \\ 6 & 100 & 1.50 \\ 3 & 120 & 1.90 \\ & & \\ \hline \end{array} $$ a. Fit a multiple regression model for predicting catch time using prey length and speed as predictors. b. Predict the catch time for an animal of prey whose length is 6 and whose speed is 50 . c. Is the multiple regression model useful for predicting catch time? Test the relevant hypotheses using \(\alpha=.05\). d. The authors of the article suggest that a simple linear regression model with the single predictor \(x=\frac{\text { length }}{\text { speed }}\) might be a better model for predicting catch time. Calculate the \(x\) values and use them to fit this linear regression model. e. Which of the two models considered (the multiple regression model from Part (a) or the simple linear regression model from Part (d)) would you recommend for predicting catch time? Justify your choice.

According to "Assessing the Validity of the PostMaterialism Index" (American Political Science Review [1999]: \(649-664\) ), one may be able to predict an individual's level of support for ecology based on demographic and ideological characteristics. The multiple regression model proposed by the authors was $$ \begin{aligned} &y=3.60-.01 x_{1}+.01 x_{2}-.07 x_{3}+.12 x_{4}+.02 x_{5} \\ &\quad-.04 x_{6}-.01 x_{7}-.04 x_{8}-.02 x_{9}+e \end{aligned} $$ where the variables are defined as follows \(y=\) ecology score (higher values indicate a greater con- $$ \begin{aligned} & \text { cern for ecology) } \\ x_{1}=& \text { age times } 10 \end{aligned} $$ \(x_{2}=\) income (in thousands of dollars) \(x_{3}=\) gender \((1=\) male, \(0=\) female \()\) \(x_{4}=\) race \((1=\) white, \(0=\) nonwhite \()\) \(x_{5}=\) education (in years) \(x_{6}=\) ideology \((4=\) conservative, \(3=\) right of center, \(2=\) middle of the road, \(1=\) left of center, and \(0=\) liberal \()\) \(x_{7}=\) social class \((4=\) upper, \(3=\) upper middle, \(2=\) middle, \(1=\) lower middle, \(0=\) lower \()\) \(x_{8}=\) postmaterialist ( 1 if postmaterialist, 0 otherwise) \(x_{9}=\) materialist \((1\) if materialist, 0 otherwise) a. Suppose you knew a person with the following characteristics: a 25-year- old, white female with a college degree (16 years of education), who has a \(\$ 32,000\) -per-year job, is from the upper middle class and considers herself left of center, but who is neither a materialist nor a postmaterialist. Predict her ecology score. b. If the woman described in Part (a) were Hispanic rather than white, how would the prediction change? c. Given that the other variables are the same, what is the estimated mean difference in ecology score for men and women? d. How would you interpret the coefficient of \(x_{2}\) ? e. Comment on the numerical coding of the ideology and social class variables. Can you suggest a better way of incorporating these two variables into the model?
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