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Chapter 14: Problem 12

A manufacturer of wood stoves collected data on \(y=\) particulate matter concentration and \(x_{1}=\) flue temperature for three different air intake settings (low, medium, and high). a. Write a model equation that includes indicator variables to incorporate intake setting, and interpret each of the \(\beta\) coefficients. b. What additional predictors would be needed to incorporate interaction between temperature and intake setting?

Short Answer

Expert verified
The model equation is \(y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} I_{m} + \beta_{3} I_{h} + e\), where \(I_{m}\) and \(I_{h}\) are indicator variables for the medium and high intake settings respectively, and \(e\) is the error term. The \(\beta\) coefficients have specific interpretations in relation to their corresponding variables. To incorporate interaction between temperature and intake setting, two additional predictors, \(x_{1}I_{m}\) and \(x_{1}I_{h}\), are needed.

Step by step solution

01

Develop a model equation to incorporate air intake setting

Let's denote the air intake setting with two indicator (dummy) variables: \(I_{m}\) for the medium intake setting, and \(I_{h}\) for the high intake setting. The low intake setting will represent the base category. Now we can write the model equation as follows: \[y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} I_{m} + \beta_{3} I_{h} + e\] where \(e\) is the error term.
02

Interpret \(\beta\) coefficients

\(\beta_{0}\) is the intercept, the expected value of \(y\) (particulate matter concentration) when the flue temperature is zero and the air intake setting is low. \(\beta_{1}\) is the increase in \(y\) for each unit increase in \(x_{1}\) (flue temperature) when the air intake setting is low. \(\beta_{2}\) is the difference in \(y\) between the medium and low air intake settings, controlling for \(x_{1}\). Likewise, \(\beta_{3}\) is the difference in \(y\) between the high and low intake settings, controlling for \(x_{1}\).
03

Identify additional predictors for interaction

To incorporate interaction between temperature (\(x_{1}\)) and intake setting, two additional predictors are needed: \(x_{1}I_{m}\) and \(x_{1}I_{h}\). These terms allow the slope of \(x_{1}\) to differ based on the level of the intake setting.

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