/*! 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 14 The article "The Value and the L... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 14

The article "The Value and the Limitations of High-Speed Turbo-Exhausters for the Removal of Tar-Fog from Carburetted Water-Gas" (Society of Chemical Industry Journal [1946]: \(166-168\) ) presented data on \(y=\) tar content \(\left(\right.\) grains \(\left./ 100 \mathrm{ft}^{3}\right)\) of a gas stream as a function of \(x_{1}=\) rotor speed (rev/minute) and \(x_{2}=\) gas inlet temperature \(\left({ }^{\circ} \mathrm{F}\right)\). A regression model using \(x_{1}, x_{2}, x_{3}=x_{2}^{2}\) and \(x_{i}=x_{1} x_{2}\) was suggested: $$ \begin{aligned} \text { mean } y \text { value }=& 86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} \\ &+.001 x_{i} \end{aligned} $$ a. According to this model, what is the mean \(y\) value if \(x_{1}=3200\) and \(x_{2}=57 ?\) b. For this particular model, does it make sense to interpret the value of a \(\beta_{2}\) as the average change in tar content associated with a 1 -degree increase in gas inlet temperature when rotor speed is held constant? Explain.

Short Answer

Expert verified
a. Insert values in the regression model to compute the mean \(y\) value. b. It wouldn't be appropriate to interpret the value of \(\beta_{2}\) as the average change in tar content with a 1-degree increase in gas inlet temperature, all else constant, because the model includes nonlinear terms, so the effect of \(x_{2}\) is dependent on its interaction with itself and other variables.

Step by step solution

01

Calculate Value of \(\overline{y}\)

Given the regression model, and values \(x_{1} = 3200\) and \(x_{2} =57\), we need to calculate \(x_{3}\) and \(x_{i}\) before we can find the mean \(y\) value. The given regression model for calculating mean \(y\) value is as follows: \(\overline{y} = 86.8-.123 x_{1}+5.09 x_{2} -.0709 x_{3} + .001 x_{i}\), where \(x_{3} = x_{2}^{2}\) and \(x_{i} = x_{1}\cdot x_{2}\).
02

Insert Values and Solve

Insert given values and solve: As some variables depend on \(x_{1}\) and \(x_{2}\), start with finding the values for \(x_{3} = 57^{2} = 3249\) and \(x_{i} = 3200 \cdot 57 = 182400\). Now substitute all these values carefully into the given model: \(\overline{y} = 86.8 - .123 \cdot 3200 +5.09 \cdot 57 - .0709 \cdot 3249 + .001 \cdot 182400\).
03

Interpretation of \(\beta_{2}\)

To interpret the value of \(\beta_{2}\), it would typically mean that for every one-unit increase in \(x_{2}\) (gas inlet temperature), the dependent variable \(y\) (tar content) would change by the value of \(\beta_{2}\), assuming all other variables are held constant. However, this model is not simple linear regression but involves both linear and quadratic terms of variables and their interaction. Here, both \(x_{2}\) and \(x_{3}\) are dependent on gas inlet temperature where \(x_{3}=x_{2}^{2}\). This indicates that the effect of temperature (both linear and squared) on tar content is not fixed but varies depending upon the value of temperature itself and its interaction with speed. Therefore, in the context of this model, interpreting \(\beta_{2}\) as the average change in tar content with a 1-degree increase in temperature, all else being constant, would not be accurate.

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Key Concepts

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

Statistical Data Analysis
Statistical data analysis involves collecting, evaluating, and summarizing numerical information to discover underlying patterns, trends, and relationships. It's a cornerstone of research that drives decision-making across various disciplines.

For instance, the exercise above uses statistical data analysis to study the effect of rotor speed and gas inlet temperature on tar content in a gas stream. By fitting these variables into a regression model, researchers can quantify how changes in these factors influence the tar content, enabling them to optimize the gas purification process.

The process begins with formulating a hypothesis, then collecting relevant data, and constructing a suitable model to test the hypothesis. The regression model created uses the data on tar content and correlates it with independent variables such as rotor speed and gas inlet temperature, including a quadratic term and an interaction term to capture more complex relationships.
Interpretation of Regression Coefficients
In regression analysis, each independent variable has an associated coefficient that estimates its relationship with the dependent variable. To interpret a regression coefficient, typically designated as \( \beta \) in statistical notation, is to understand the expected change in the dependent variable for a one-unit increase in the independent variable, while keeping all other variables constant.

Consider the coefficient \( \beta_{2} \) from the exercise, associated with the gas inlet temperature (\( x_{2} \) variable). Under simple linear regression assumptions, \( \beta_{2} \) would represent the average change in the tar content for every one-degree rise in temperature. However, in a multiple regression context, especially with higher-order terms and interaction terms, the interpretation becomes more nuanced because the effect of changing \( x_{2} \) could be different at various levels of \( x_{2} \) itself or in combination with other variables such as rotor speed. Therefore, the interpretation of \( \beta_{2} \) here would not purely indicate the isolated effect of temperature on tar content.
Multiple Regression
Multiple regression is an extension of simple linear regression that includes two or more predictor variables. It helps to model the relationship between a dependent variable and several independent variables, capturing more complexity than a simple linear relationship.

The exercise proposes a multiple regression model with variables \( x_{1}, x_{2}, x_{3} \) (where \( x_{3} = x_{2}^{2} \) represents the squared temperature), and \( x_{i} \) (representing the interaction between \( x_{1} \) and \( x_{2} \) 鈥� rotor speed and gas inlet temperature). The inclusion of squared and interaction terms allows the model to account for non-linear relationships and the combined effect of two variables affecting the outcome, respectively.

Applying a multiple regression model helps researchers and analysts understand and predict complex phenomena by factoring in multiple variables together, leading to more accurate and realistic models compared to a single variable regression analysis. The model in the exercise, hence, provides a richer and more detailed understanding of the factors influencing tar content in the gas stream.

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Most popular questions from this chapter

The authors of the paper "Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thidnness. Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International journal of Poultry Science [2008]: \(85-88\) ) used a multiple regression model with two independent variables where $$ \begin{aligned} y &=\text { quail egg weight }(\mathrm{g}) \\ x_{1} &=\text { egg width }(\mathrm{mm}) \\ x_{2} &=\text { egg length }(\mathrm{mm}) \end{aligned} $$ The regression function suggested in the paper is \(-21.658+0.828 x_{1}+0.373 x_{2}\) a. What is the mean egg weight for quail eggs that have a width of \(20 \mathrm{~mm}\) and a length of \(50 \mathrm{~mm}\) ? b. Interpret the values of \(\beta_{1}\) and \(\beta_{2}\).

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 \(x_{i}=\) process temperature ("Advantages of CE-HDP Bleaching for High Brightness Kraft Pulp Production." TAPPI [1964]: \(107 \mathrm{~A}-173 \mathrm{~A}\) ). $$ \begin{array}{ccccc} x_{1} & x_{2} & x_{3} & x_{4} & y \\ \hline .2 & .2 & 1.5 & 145 & 83.9 \\ .4 & .2 & 1.5 & 145 & 84.9 \\ .2 & .4 & 1.5 & 145 & 83.4 \\ .4 & .4 & 1.5 & 145 & 84.2 \\ .2 & .2 & 3.5 & 145 & 83.8 \\ .4 & .2 & 3.5 & 145 & 84.7 \\ .2 & .4 & 3.5 & 145 & 84.0 \\ .4 & .4 & 3.5 & 145 & 84.8 \\ .2 & .2 & 1.5 & 175 & 84.5 \\ .4 & .2 & 1.5 & 175 & 86.0 \\ .2 & .4 & 1.5 & 175 & 82.6 \\ .4 & .4 & 1.5 & 175 & 85.1 \\ .2 & .2 & 3.5 & 175 & 84.5 \\ .4 & .2 & 3.5 & 175 & 86.0 \\ .2 & .4 & 3.5 & 175 & 84.0 \\ .4 & .4 & 3.5 & 175 & 85.4 \\ .1 & .3 & 2.5 & 160 & 82.9 \\ .5 & .3 & 2.5 & 160 & 85.5 \\ .3 & .1 & 2.5 & 160 & 85.2 \\ .3 & .5 & 2.5 & 160 & 84.5 \\ .3 & .3 & 0.5 & 160 & 84.7 \\ .3 & .3 & 4.5 & 160 & 85.0 \end{array} $$ $$ \begin{array}{ccccc} x_{1} & x_{2} & x_{3} & x_{4} & y \\ \hline .3 & .3 & 2.5 & 130 & 84.9 \\ .3 & .3 & 2.5 & 190 & 84.0 \\ .3 & .3 & 2.5 & 160 & 84.5 \\ .3 & .3 & 2.5 & 160 & 84.7 \\ .3 & .3 & 2.5 & 160 & 84.6 \\ .3 & .3 & 2.5 & 160 & 84.9 \\ .3 & .3 & 2.5 & 160 & 84.9 \\ .3 & .3 & 2.5 & 160 & 84.5 \\ .3 & .3 & 2.5 & 160 & 84.6 \\ \hline \end{array} $$ 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 \(s_{c}\).

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 in which 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. $$ \begin{array}{ccc} \text { Prey Length } & \text { Prey Speed } & \text { Catch Time } \\ \hline 7 & 20 & 1.10 \\ 6 & 20 & 1.20 \\ 5 & 20 & 1.23 \\ 4 & 20 & 1.40 \\ 3 & 20 & 1.50 \\ 3 & 40 & 1.40 \\ 4 & 40 & 1.36 \\ 6 & 40 & 1.30 \\ 7 & 40 & 1.28 \\ 7 & 80 & 1.40 \\ 6 & 60 & 1.38 \\ 5 & 80 & 1.40 \\ 7 & 100 & 1.43 \\ 6 & 100 & 1.43 \\ 7 & 120 & 1.70 \\ 5 & 80 & 1.50 \\ 3 & 80 & 1.40 \\ 6 & 100 & 1.50 \\ 3 & 120 & 1.90 \\ \hline \end{array} $$ 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 Post-Materialism 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 $$ \begin{aligned} y=& 3.60-.01 x_{1}+.01 x_{2}-.07 x_{3}+.12 x_{6}+.02 x_{5} \\ &-.04 x_{6}-.01 x_{7}-.04 x_{8}-.02 x_{9}+e \end{aligned} $$ where the variables are defined as follows: \(y=\) ecology score (higher values indicate a greater concern for ecology) \(x_{1}=\) age times 10 \(x_{2}=\) income (in thousands of dollars) \(x_{3}=\) gender \((1=\) male, \(0=\) female \()\) \(x_{j}=\operatorname{race}(1=\) 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 \(\begin{aligned}\text { center, and } 0=\text { liberal }) \\ x_{7}=& \text { social class }(4=\text { upper, } 3=\text { upper middle, }\\\ 2=\text { middle, } 1=\text { lower middle, and } \\\0=\text { lower }) \end{aligned}\) \(x_{8}=\) 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 $$\$ 32,000$$ -peryear 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?

Obtain as much information as you can about the \(P\) -value for the \(F\) test for model utility in each of the following situations: a. \(k=2, n=21\), calculated \(F=2.47\) b. \(k=8, n=25\), calculated \(F=5.98\) c. \(k=5, n=26\), calculated \(F=3.00\) d. The full quadratic model based on \(x_{1}\) and \(x_{2}\) is fit, \(n=20\), and calculated \(F=8.25\). e. \(k=5, n=100\), calculated \(F=2.33\)
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