/*! 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 23 A quality control engineer is in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 23

A quality control engineer is interested in predicting the strength of particle board \(y\) as a function of the size of the particles \(x_{1}\) and two types of bonding compounds. If the basic response is expected to be a quadratic function of particle size, write a linear model that incorporates the qualitative variable "bonding compound" into the predictor equation.

Short Answer

Expert verified
Answer: To create a linear regression model that predicts the strength of particle board, we will write a quadratic function of particle size with an indicator variable to account for the effect of two types of bonding compounds. The linear model can be expressed as \(y = \beta_{0}+\beta_{1}x_{1}+\beta_{2}x^2_{1}+\beta_{3}I_{1}+\epsilon\).

Step by step solution

01

Define the Indicator Variable for Bonding Compounds

Let's define an indicator variable \(I_{1}\) for bonding compound types: $I_{1} = \begin{cases} 1, &\mbox{if bonding compound type is Compound 1}\\ 0, &\mbox{if bonding compound type is Compound 2} \end{cases}$
02

Write the quadratic function of particle size

Since the basic response is expected to be a quadratic function of particle size, let's write it: \(y = \beta_{0}+\beta_{1}x_{1}+\beta_{2}x^2_{1}+\epsilon\)
03

Incorporate the indicator variable for bonding compound types into the model

Now, we will incorporate the indicator variable \(I_{1}\) into the linear model to account for the effect of the two different types of bonding compounds: \(y = \beta_{0}+\beta_{1}x_{1}+\beta_{2}x^2_{1}+\beta_{3}I_{1}+\epsilon\) Here, - \(\beta_{0}\) is the intercept, - \(\beta_{1}\) and \(\beta_{2}\) are the coefficients for the linear and quadratic terms of particle size \(x_{1}\), respectively, - \(\beta_{3}\) represents the difference in the response due to Compound 1 compared to Compound 2, - \(I_{1}\) is the indicator variable for bonding compound types, and - \(\epsilon\) is the error term. Now, we have a linear model that incorporates the qualitative variable "bonding compound" into the predictor equation. This model can be used for predicting the strength of particle board \(y\) as a function of the size of the particles \(x_{1}\) and two types of bonding compounds.

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.

Indicator Variables
An indicator variable, also referred to as a dummy variable, is used in statistical models to represent categorical data, such as gender, race, or, as in our exercise, types of bonding compounds. In linear regression models, we incorporate indicator variables to take into account the impact of qualitative variables.

For instance, if we're dealing with two categories of bonding compounds, we can create an indicator variable, let's call it \(I_{1}\), which takes the value of 1 if the particle board uses Compound 1, and 0 if it uses Compound 2. This binary system allows the model to adjust the predicted value of \(y\) depending on which compound is used, accounting for the fact that different compounds might have different effects on the strength of the particle board.
Quadratic Function
In the context of regression models, a quadratic function is an equation that involves a predictor variable raised to the second power, effectively capturing the curvature in the relationship between the predictor and the response variable. The general form of a quadratic function is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the predictor variable.

In our exercise, the strength of the particle board (\(y\)) is expected to be a quadratic function of the particle size (\(x_{1}\)). This means that the relationship between particle size and board strength isn't just linear; there's a curvature indicating that increases in particle size have a non-linear effect on the board's strength.
Predictor Equation
The predictor equation in a linear regression model is the equation used to predict the value of the dependent variable based on the values of the independent variables. It is composed of the intercept, the coefficients for each predictor, and the predictor variables themselves.

Incorporating a quadratic function and an indicator variable as shown in our exercise produces the predictor equation \(y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{1}^2 + \beta_{3}I_{1} + \epsilon\). This equation allows us to predict the strength of particle board by considering both the size of the particles and the type of bonding compound used. The term \(\epsilon\) represents the error term, capturing the deviation of the observed values from the values predicted by the model.
Qualitative Variables
Qualitative variables, unlike quantitative variables, categorize or describe attributes or properties without a natural sense of order or magnitude. Examples include brands, colors, or types of a material such as the bonding compounds in our exercise.

To include qualitative variables in a regression model, we use indicator variables that act as proxies. They allow us to assign numeric values to the categories, which can then be incorporated into the regression model, enabling it to distinguish between groups. By doing so, we can examine the influence of non-quantitative factors on the dependent variable鈥攊n this case, the strength of a particle board.

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

Because dolphins (and other large marine mammals) are considered to be the top predators in the marine food chain, the heavy metal concentrations in striped dolphins were measured as part of a marine pollution study. The concentration of mercury, the heavy metal reported in this study, is expected to differ in males and females because the mercury in a female is apparently transferred to her offspring during gestation and nursing. This study involved 28 males between the ages of .21 and 39.5 years, and 17 females between the ages of .80 and 34.5 years. For the data in the table, $$ \begin{aligned} x_{1}=& \text { Age of the dolphin (in years) } \\ x_{2}=&\left\\{\begin{array}{ll} 0 & \text { if female } \\ 1 & \text { if male } \end{array}\right.\\\ y=& \text { Mercury concentration (in } \\ & \text { micrograms/gram) in the liver } \end{aligned} $$ $$ \begin{array}{rrl|lll} y & x_{1} & x_{2} & y & x_{1} & x_{2} \\ \hline 1.70 & .21 & 1 & 481.00 & 22.50 & 1 \\ 1.72 & .33 & 1 & 485.00 & 24.50 & 1 \\ 8.80 & 2.00 & 1 & 221.00 & 24.50 & 1 \\ 5.90 & 2.20 & 1 & 406.00 & 25.50 & 1 \\ 101.00 & 8.50 & 1 & 252.00 & 26.50 & 1 \\ 85.40 & 11.50 & 1 & 329.00 & 26.50 & 1 \\ 118.00 & 11.50 & 1 & 316.00 & 26.50 & 1 \\ 183.00 & 13.50 & 1 & 445.00 & 26.50 & 1 \\ 168.00 & 16.50 & 1 & 278.00 & 27.50 & 1 \\ 218.00 & 16.50 & 1 & 286.00 & 28.50 & 1 \\ 180.00 & 17.50 & 1 & 315.00 & 29.50 & 1 \\ 264.00 & 20.50 & 1 & & & \end{array} $$ $$ \begin{array}{rrl|lll} y & x_{1} & x_{2} & y & x_{1} & x_{2} \\ \hline 241.00 & 31.50 & 1 & 142.00 & 17.50 & 0 \\ 397.00 & 31.50 & 1 & 180.00 & 17.50 & 0 \\ 209.00 & 36.50 & 1 & 174.00 & 18.50 & 0 \\ 314.00 & 37.50 & 1 & 247.00 & 19.50 & 0 \\ 318.00 & 39.50 & 1 & 223.00 & 21.50 & 0 \\ 2.50 & .80 & 0 & 167.00 & 21.50 & 0 \\ 9.35 & 1.58 & 0 & 157.00 & 25.50 & 0 \\ 4.01 & 1.75 & 0 & 177.00 & 25.50 & 0 \\ 29.80 & 5.50 & 0 & 475.00 & 32.50 & 0 \\ 45.30 & 7.50 & 0 & 342.00 & 34.50 & 0 \\ 101.00 & 8.05 & 0 & & & \\ 135.00 & 11.50 & 0 & & & \end{array} $$ a. Write a second-order model relating \(y\) to \(x_{1}\) and \(x_{2}\). Allow for curvature in the relationship between age and mercury concentration, and allow for an interaction between gender and age. Use a computer software package to perform the multiple regression analysis. Refer to the printout to answer these questions. b. Comment on the fit of the model, using relevant statistics from the printout. c. What is the prediction equation for predicting the mercury concentration in a female dolphin as a function of her age? d. What is the prediction equation for predicting the mercury concentration in a male dolphin as a function of his age? e. Does the quadratic term in the prediction equation for females contribute significantly to the prediction of the mercury concentration in a female dolphin? f. Are there any other important conclusions that you feel were not considered regarding the fitted prediction equation?

Suppose you wish to predict production yield \(y\) as a function of several independent predictor variables. Indicate whether each of the following independent variables is qualitative or quantitative. If qualitative, define the appropriate dummy variable(s). a. The prevailing interest rate in the area b. The price per pound of one item used in the production process c. The plant (A, B, or C) at which the production yield is measured d. The length of time that the production machine has been in operation e. The shift (night or day) in which the yield is measured

Suppose that you fit the model $$ E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3} $$ to 15 data points and found \(F\) equal to 57.44 . a. Do the data provide sufficient evidence to indicate that the model contributes information for the prediction of \(y\) ? Test using a \(5 \%\) level of significance. b. Use the value of \(F\) to calculate \(R^{2}\). Interpret its value.

A department store conducted an experiment to investigate the effects of advertising expenditures on the weekly sales for its men's wear, children's wear, and women's wear departments. Five weeks for observation were randomly selected from each department, and an advertising budget \(x_{1}\) (in hundreds of dollars) was assigned for each. The weekly sales (in thousands of dollars) are shown in the accompanying table for each of the 15 one-week sales periods. If we expect weekly sales \(E(y)\) to be linearly related to advertising expenditure \(x_{1},\) and if we expect the slopes of the lines corresponding to the three departments to differ, then an appropriate model for \(E(y)\) is where $$ \begin{array}{l} x_{1}=\text { Advertising expenditure } \\ x_{2}=\left\\{\begin{array}{ll} 1 & \text { if children's wear department B } \\ 0 & \text { if not } \end{array}\right. \\ x_{3}=\left\\{\begin{array}{ll} 1 & \text { if women's wear department } \mathrm{C} \\ 0 & \text { if not } \end{array}\right. \end{array} $$ a. Find the equation of the line relating \(E(y)\) to advertising expenditure \(x_{1}\) for the men's wear department A. [HINT: According to the coding used for the dummy variables, the model represents mean sales \(E(y)\) for the men's wear department \(\mathrm{A}\) when \(x_{2}=x_{3}=0 .\) Substitute \(x_{2}=x_{3}=0\) into the equation for \(E(y)\) to find the equation of this line. b. Find the equation of the line relating \(E(y)\) to \(x_{1}\) for the children's wear department B. [HINT: According to the coding, the model represents \(E(y)\) for the children's wear department when \(x_{2}=1\) and \(\left.x_{3}=0 .\right]\) c. Find the equation of the line relating \(E(y)\) to \(x_{1}\) for the women's wear department \(\mathrm{C}\). d. Find the difference between the intercepts of the \(E(y)\) lines corresponding to the children's wear \(\mathrm{B}\) and men's wear A departments. e. Find the difference in slopes between \(E(y)\) lines corresponding to the women's wear \(\mathrm{C}\) and men's wear A departments. f. Refer to part e. Suppose you want to test the null hypothesis that the slopes of the lines corresponding to the three departments are equal. Express this as a test of hypothesis about one or more of the model parameters.

Until recently, performance tires were fitted mostly on sporty or luxury vehicles. Now they come standard on many everyday sedans. Increased levels of handling and grip have come at the expense of tread wear. The data that follows is abstracted from a report on all-season tires by Consumer Reports \(^{8}\) in which several aspects of performance were evaluated for \(n=26\) different tires where $$ \begin{aligned} y &=\text { overall score } & x_{1} &=\text { dry braking } \\ x_{2} &=\text { wet braking } & x_{3} &=\text { handling } \\ x_{4} &=\text { roll resistance } & x_{5} &=\text { tread life } \end{aligned} $$ $$ \begin{array}{lrrrrrrr} \text { Brand and Model } & \text { Price (\$) } & y & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ \hline \begin{array}{l} \text { Michelin HydroEdge } \\ \text { Continental ProContact } \end{array} & 116 & 84 & 4 & 5 & 4 & 3 & 5 \\ \text { ECOPLUS } & 90 & 82 & 4 & 4 & 3 & 5 & 3 \\ \text { Michelin Energy Saver A/S } & 120 & 82 & 4 & 4 & 3 & 5 & 3 \end{array} $$ $$ \begin{array}{lrrrrrrr} \text { Brand and Model } & \text { Price (\$) } & y & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ \hline \text { Hankook 0ptimo H727 } & 96 & 82 & 4 & 4 & 3 & 3 & 3 \\ \text { Pirelli P4 Four Seasons } & 98 & 80 & 4 & 5 & 3 & 2 & 2 \\ \text { Goodyear Assurance TripleTred } & 121 & 80 & 4 & 4 & 3 & 3 & 3 \\ \text { Pirelli Cinturato P5 } & 98 & 78 & 3 & 5 & 3 & 2 & 3 \\ \text { Kumho Solus KR21 } & 77 & 78 & 4 & 5 & 3 & 3 & 4 \\ \text { Maxxis Escapade MA-T1 } & 77 & 76 & 3 & 4 & 4 & 3 & 4 \\ \text { Toyo Extensa A/S } & 99 & 76 & 3 & 4 & 3 & 3 & 3 \\ \text { Cooper GFE } & 83 & 76 & 3 & 4 & 3 & 4 & 2 \\ \text { Toyo Versado LX } & 91 & 76 & 4 & 4 & 3 & 3 & 5 \\ \text { BFGoodrich Traction T/A T } & 90 & 74 & 4 & 4 & 4 & 2 & 3 \\ \text { General Altimax RT } & 77 & 74 & 3 & 5 & 3 & 3 & 3 \\ \text { Yokohama Avid TRZ } & 81 & 74 & 4 & 5 & 3 & 2 & 3 \\ \text { Dayton Quadra LE } & 74 & 74 & 3 & 4 & 3 & 3 & 3 \\ \text { Cooper CS4 Touring } & 86 & 72 & 3 & 5 & 3 & 2 & 3 \\ \text { Uniroyal Tiger Paw Tour SR } & 77 & 72 & 4 & 4 & 2 & 5 & 2 \\ \text { Yokohama Avid Touring-S } & 70 & 70 & 4 & 4 & 3 & 4 & 4 \\ \text { Cooper Lifeliner GLS } & 78 & 68 & 3 & 4 & 3 & 4 & 2 \\ \text { Yokohama Avid T4 } & 85 & 66 & 3 & 4 & 3 & 3 & 2 \\ \text { Bridgestone Turanze EL400 } & 97 & 66 & 3 & 4 & 2 & 3 & 2 \\ \text { Falken Sincera SN828 } & 82 & 64 & 3 & 5 & 3 & 3 & 1 \\ \text { Dunlop SP 60 } & 79 & 64 & 3 & 3 & 2 & 2 & 4 \\ \text { Sumitomo HTR T4 } & 67 & 64 & 3 & 4 & 3 & 4 & 2 \\ \text { Firestone FR710 } & 80 & 60 & 3 & 5 & 3 & 3 & 4 \end{array} $$ a. Use a program of your choice to find the correlation matrix for the variables under study including price. Is price significantly correlated with any of the study variables? Which variables appear to be highly correlated with \(y,\) the overall score? b. Write a model to describe \(y,\) overall score, as a function of the variables \(x_{1}=\) dry braking, \(x_{2}=\) wet braking \(, x_{3}=\) handling, \(x_{4}=\) roll resistance, and \(x_{5}=\) tread life. c. Use a regression program of your choice to fit the full model using all of the predictors. What proportion of the variation in \(y\) is explained by regression? Does this convey the impression that the model adequately explains the inherent variability in \(y ?\) d. Which variable or variables appear to be good predictors of \(y\) ? How might you refine the model in light of these results? Use these variables in refitting the model. What proportion of the variation is explained by this refitted model? Comment on the adequacy of this reduced model in comparison to the full model.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.