/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 The article "Pulp Brightness Rev... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 9

The article "Pulp Brightness Reversion: Influence of Residual Lignin on the Brightness Reversion of Bleached Sulfite and Kraft Pulps鈥� (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 population regression 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 equation would show a curve with corresponding y-values for x-values of 2, 4, 6, 8, 10, and 12. For a degree of delignification value of 8 and 10, the mean chlorine content would depend on their respective \(y\) values. The change in mean chlorine content when the degree of delignification increases sequentially would be calculated by subtracting relevant \(y\) values.

Step by step solution

01

Substituting Values for Graph

Start by substituting the given \(x\) values into the regression equation to get the corresponding \(y = 220 + 75x - 4x^2\). Therefore, you calculate the \(y\) values when \(x = 2, 4, 6, 8, 10, 12\). Afterwards, these coordinates are plotted on a graph to sketch the regression model.
02

Comparing Chlorine Content

Substitute \(x = 8\) and \(x = 10\) into the regression equation separately. The resulting \(y\) values represent mean chlorine content. Compare the results to determine which \(x\) value produces a higher mean chlorine content.
03

Calculating Change in Chlorine Content

To determine the change in mean chlorine content, calculate the \(y\) values for \(x = 8, 9, 10\). The change when \(x\) increases from 8 to 9 is \(y_9 - y_8\). Similarly, the change when \(x\) increases from 9 to 10 is \(y_{10} - y_9\).

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 91影视!

Key Concepts

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

Regression Equation
A regression equation is used in statistics to describe the relationship between one dependent variable and one or more independent variables. The quadratic regression model, like the one given in the exercise with the equation y = 220 + 75x - 4x^2 + e, is a type of multiple regression that includes squared terms of the independent variable.
This allows the model to capture curvature in the relationship between the variables, which can be particularly useful when studying phenomena such as the brightness of paper pulp that do not follow a straight-line pattern. Quadratic models can reveal maximum and minimum points, which might correspond to optimal operating conditions in industrial processes.
When constructing a graph for a quadratic regression model, you plot several points by substituting different values of x (the independent variable) and calculate the corresponding values of y (the dependent variable). Connect these points with a smooth curve to visualize the trend, which, in a quadratic model, will typically form a parabola. Interpreting this graph can help predict outcomes and understand the effect of varying the independent variable, such as the degree of delignification in paper processing.
Degree of Delignification
The degree of delignification refers to the extent to which lignin, a natural component that binds plant cells together, is removed from wood pulp during the production of paper. Delignification is key in paper manufacturing because it affects both the strength and color of the paper. Lignin tends to darken paper, so its removal is essential for producing brighter paper.
In the context of the exercise, the degree of delignification is represented by the independent variable x. As x increases, it indicates a greater level of lignin removal. The regression equation allows us to determine how this increase in delignification affects the chlorine content used in the bleaching process. Chlorine is traditionally used to break down lignin, but excessive chlorine can lead to environmental issues and is also costly.
Understanding the optimal degree of delignification helps manufacturers balance the trade-off between paper brightness and manufacturing costs, as well as environmental considerations. The regression model therefore aids in pinpointing this balance by predicting chlorine content needed at different stages of lignin removal.
Mean Chlorine Content
The mean chlorine content is a critical variable in the paper bleaching process. It represents the average amount of chlorine used to treat the pulp in order to reach a desired level of delignification. Chlorine reacts with lignin, removing it from the pulp, which in turn increases the paper's brightness.
From the given quadratic model, we can determine the mean chlorine content for different levels of delignification by substituting those levels into the equation. For instance, as the exercise suggests, determining whether the mean chlorine content is higher for a degree of delignification of 8 or 10 is as simple as computing and comparing the respective y values for each x. This information is directly related to both the efficiency and the environmental impact of the bleaching process. Suppliers aim to optimize the mean chlorine content to avoid using too much chlorine鈥攚hich can save costs and reduce environmental harm鈥攚hile still achieving the necessary level of paper whiteness.
Moreover, by calculating the incremental changes in mean chlorine content as the degree of delignification increases (e.g., from 8 to 9, or from 9 to 10), manufacturers can understand the marginal impact of delignification on resource utilization, guiding them in process optimization and control.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Data from a sample of \(n=150\) quail eggs were used to fit a multiple regression model relating $$ y=\text { eggshell surface area }\left(\mathrm{mm}^{2}\right) $$ \(x_{1}=\) egg weight \((\mathrm{g})\) \(x_{2}=e g g\) width \((\mathrm{mm})\) $$ x_{3}=\text { egg length }(\mathrm{mm}) $$ (鈥淧redicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors," International Journal of Poultry Science [2008]: 85-88). The resulting estimated regression function was $$ \begin{array}{l} 10.561+1.535 x_{1}-0.178 x_{2}-0.045 x_{3} \\ \text { and } R^{2}=.996 \end{array} $$ a. Carry out a model utility test to determine if this multiple regression model is useful. b. A simple linear regression model was also used to describe the relationship between \(y\) and \(x_{1}\), resulting in the estimated regression function \(6.254+1.387 x_{1}\). The \(P\) -value for the associated model utility test was reported to be less than .01 , and \(r^{2}=.994 .\) Is the linear model useful? Explain. c. Based on your answers to Parts (a) and (b), which of the two models would you recommend for predicting eggshell surface area? Explain the rationale for your choice.

The following statement appeared in the article 鈥淒imensions of Adjustment Among College Women鈥� (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 \(\begin{aligned} y &=\text { current GPA } \\ x_{1} &=\text { academic adjustment score } \\ x_{2} &=\text { race }(\text { with white }=0, \text { other }=1) \end{aligned}\) What multiple regression model is suggested by the statement? Did you include an interaction term in the model? Why or why not?

This exercise requires the use of a computer package. The accompanying data resulted from a study of the relationship between \(y=\) brightness of finished paper and the independent variables \(x_{1}=\) hydrogen peroxide \((\%\) by weight), \(x_{2}=\) sodium hydroxide (\% by weight), \(x_{3}=\) silicate (\% by weight), and \(x_{4}=\) process temperature (鈥淎dvantages of CE-HDP Bleaching for High Brightness Kraft Pulp Production," TAPPI [1964]: \(107 \mathrm{~A}-173 \mathrm{~A})\). a. Find the estimated regression equation for the model that includes all independent variables, all quadratic terms, and all interaction terms. b. Using a .05 significance level, perform the model utility test. c. Interpret the values of the following quantities: SSResid, \(R^{2},\) and \(s_{e}\)

When coastal power stations take in large quantities of cooling water, it is inevitable that a number of fish are drawn in with the water. Various methods have been designed to screen out the fish. The article 鈥淢ultiple Regression Analysis for Forecasting Critical Fish Influxes at Power Station Intakes" (Journal of Applied Ecology [1983]: 33-42) examined intake fish catch at an English power plant and several other variables thought to affect fish intake: \(\begin{aligned} y &=\text { fish intake (number of fish) } \\ x_{1} &=\text { water temperature }\left({ }^{\circ} \mathrm{C}\right) \\ x_{2} &=\text { number of pumps running } \\ x_{3} &=\text { sea state }(\text { values } 0,1,2, \text { or } 3) \\ x_{4} &=\text { speed }(\mathrm{knots}) \end{aligned}\) Part of the data given in the article were used to obtain the estimated regression equation $$ \hat{y}=92-2.18 x_{1}-19.20 x_{2}-9.38 x_{3}+2.32 x_{4} $$ (based on \(n=26\) ). SSRegr \(=1486.9\) and SSResid = 2230.2 were also calculated. a. Interpret the values of \(b_{1}\) and \(b_{4}\) b. What proportion of observed variation in fish intake can be explained by the model relationship? c. Estimate the value of \(\sigma\). d. Calculate adjusted \(R^{2} .\) How does it compare to \(R^{2}\) itself?

The relationship between yield of maize, date of planting, and planting density was investigated in the article 鈥淒evelopment of a Model for Use in Maize Replant Decisions鈥� (Agronomy Journal [1980]: \(459-464)\). Let $$ \begin{aligned} y &=\text { percent maize yield } \\ x_{1} &=\text { planting date }(\text { days after } \text { April } 20) \\ x_{2} &=\text { planting density }(10,000 \text { plants } / \mathrm{ha}) \end{aligned} $$ The regression model with both quadratic terms \((y=\alpha\) $$ +\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+e \text { where } x_{3}=x_{1}^{2} $$ and \(x_{4}=x_{2}^{2}\) ) provides a good description of the relationship between \(y\) and the independent variables. a. If \(\alpha=21.09, \beta_{1}=.653, \beta_{2}=.0022, \beta_{3}=2.0206\), and \(\beta_{4}=0.4\), what is the population regression function? b. Use the regression function in Part (a) to determine the mean yield for a plot planted on May 6 with a density of 41,180 plants/ha. c. Would the mean yield be higher for a planting date of May 6 or May 22 (for the same density)? d. Is it legitimate to interpret \(\beta_{1}=.653\) as the average change in yield when planting date increases by one day and the values of the other three predictors are held fixed? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.