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Chapter 14: Problem 13

Consider a regression analysis with three independent variables \(x_{1}, x_{2}\), and \(x_{3}\). Give the equation for the following regression models: a. The model that includes as predictors all independent variables but no quadratic or interaction terms; b. The model that includes as predictors all independent variables and all quadratic terms; c. All models that include as predictors all independent variables, no quadratic terms, and exactly one interaction term; d. The model that includes as predictors all independent variables, all quadratic terms, and all interaction terms (the full quadratic model).

Short Answer

Expert verified
a. Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3\)\n b. Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2\)\n c.\n - Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1x_2\)\n - Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_5x_1x_3\)\n - Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_6x_2x_3\)\n d. Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2 + \beta_7x_1x_2 + \beta_8x_1x_3 + \beta_9x_2x_3\)

Step by step solution

01

a. All independent variables, no quadratic or interaction terms

This model includes each of the independent variables, \(x_1\), \(x_2\), and \(x_3\), but does not include any quadratic or interaction terms. The equation for this model is as follows: Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3\)
02

b. All independent variables and all quadratic terms

This model includes each of the independent variables, \(x_1\), \(x_2\), and \(x_3\), and their quadratic terms, but no interaction terms. The equation for this model is as follows: Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2\)
03

c. All independent variables, no quadratic terms, and exactly one interaction term

This model includes each of the independent variables, \(x_1\), \(x_2\), and \(x_3\), and one interaction term, but no quadratic terms. Since we have three variables, we end up with three different models. The equations for these models are as follows: Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1x_2\) Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_5x_1x_3\) Y = \(\beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_6x_2x_3\)
04

d. Full quadratic model

The full quadratic model includes each of the independent variables, \(x_1\), \(x_2\), and \(x_3\), their quadratic terms, and all interaction terms. The equation for this model is as follows: \[ Y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2 + \beta_7x_1x_2 + \beta_8x_1x_3 + \beta_9x_2x_3 \]

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Most popular questions from this chapter

The accompanying Minitab output results from fitting the model described in Exercise \(14.14\) to data. $$ \begin{aligned} &\begin{array}{lrrr} \text { Predictor } & \text { Coet } & \text { Stdev } & \text { t-ratio } \\ \text { Constant } & 86.85 & 85.39 & 1.02 \\ \text { X1 } & -0.12297 & 0.03276 & -3.75 \\ \text { X2 } & 5.090 & 1.969 & 2.58 \\ \text { X3 } & -0.07092 & 0.01799 & -3.94 \\ \text { X }_{4} & 0.0015380 & 0.0005560 & 2.77 \\ \mathrm{~S}=4.784 & \text { R-sq }=90.8 \% & R-\mathrm{sq}(\mathrm{adj})=89.4 \% \end{array}\\\ &\text { Analysis of Variance }\\\ &\begin{array}{lrrr} & \text { DF } & \text { SS } & \text { MS } \\ \text { Regression } & 4 & 5896.6 & 1474.2 \\ \text { Error } & 26 & 595.1 & 22.9 \\ \text { Total } & 30 & 6491.7 & \end{array} \end{aligned} $$ a. What is the estimated regression equation? b. Using a \(.01\) significance level, perform the model utility test. c. Interpret the values of \(R^{2}\) and \(s_{e}\) given in the output.

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 \(s_{c}\).

A number of investigations have focused on the problem of assessing loads that can be manually handled in a safe manner. The article "Anthropometric, Musde Strength, and Spinal Mobility Characteristics as Predictors in the Rating of Acceptable Loads in Parcel Sorting" (Ergonomics [1992]: 1033-1044) proposed using a regression model to relate the dependent variable \(y=\) individual's rating of acceptable load \((\mathrm{kg})\) to \(k=3\) independent (predictor) variables: $$ \begin{aligned} &x_{1}=\text { extent of left lateral bending }(\mathrm{cm}) \\ &x_{2}=\text { dynamic hand grip endurance (seconds) } \\ &x_{3}=\text { trunk extension ratio }(\mathrm{N} / \mathrm{kg}) \end{aligned} $$ Suppose that the model equation is $$ y=30+.90 x_{1}+.08 x_{2}-4.50 x_{3}+e $$ and that \(\sigma=5\). a. What is the population regression function? b. What are the values of the population regression coefficients? c. Interpret the value of \(\beta_{1}\). d. Interpret the value of \(\beta_{3}\). e. What is the mean rating of acceptable load when extent of left lateral bending is \(25 \mathrm{~cm}\), dynamic hand grip endurance is 200 seconds, and trunk extension ratio is \(10 \mathrm{~N} / \mathrm{kg}\) ? f. If repeated observations on rating are made on different individuals, all of whom have the values of \(x_{1}\), \(x_{2}\), and \(x_{3}\) specified in Part (e), in the long run approximately what percentage of ratings will be between \(13.5 \mathrm{~kg}\) and \(33.5 \mathrm{~kg}\) ?

Suppose that a multiple regression data set consists of \(n=15\) observations. For what values of \(k\), the number of model predictors, would the corresponding model with \(R^{2}=.90\) be judged useful at significance level .05? Does such a large \(R^{2}\) value necessarily imply a useful model? Explain.
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