Question

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 \(y=\operatorname{tar}\) content (grains/100 \(\mathrm{ft}^{3}\) ) 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 \(x_{4}=x_{1} x_{2}\) was suggested: $$ \text { mean } y \text { value }=86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} $$ \(+.001 x_{4}\) a. According to this model, what is the mean \(y\) value if $$ x_{1}=3200 \text { and } x_{2}=57 ? $$ 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

a. A specific formula needs to be followed to derive it. b. It doesn't make sense to interpret the coefficient \( \beta_{2} \) as the average change in tar content associated with a 1-degree increase in gas inlet temperature when rotor speed is held constant, due to the interaction and quadratic term in the regression model.

Step-by-step solution

Calculate the Mean Y Value

First calculate the values of \( x_{3} \) and \( x_{4} \). Given \( x_{1}=3200 \) and \( x_{2}=57 \), \( x_{3}=x_{2}^{2}=(57)^2 = 3249 \) and \( x_{4}=x_{1} x_{2}=3200*57 = 182400 \). Now, substitute these values into the regression model \( y=86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} +.001 x_{4} \) and calculate the result, which gives the mean y value.

Interpret the Coefficient \( \beta_{2} \)

Generally, in a simple linear regression, \( \beta_{2} \) can be interpreted as the average change in the output variable \( y \) for a one-unit change in \( x_{2} \), with other factors being held constant. However, in this case, our model includes an interaction term, \( x_{2}x_{1} \), and a quadratic term, \( x_{2}^2 \), which means the effect of \( x_{2} \) on \( y \) is not constant, but varies depending on the values of \( x_{1} \) and \( x_{2} \). Thus, the interpretation of \( \beta_{2} \) as the average change in tar content associated with a 1-degree increase in gas inlet temperature when rotor speed is held constant does not hold.

Key concepts

Statistical Regression

Statistical regression is a powerful tool used to examine the relationship between one dependent variable and one or more independent variables. By fitting a regression line through the data points, we can predict the mean value of the dependent variable based on the given values of the independent variables. The coefficients in a regression model, denoted as beta values \( \beta \), represent how much the dependent variable is expected to change with a one-unit change in the respective independent variable, all else being constant.

For example, in the exercise involving the removal of tar-fog from carburetted water-gas, the regression model considers the tar content as the dependent variable and rotor speed, gas inlet temperature and their derivatives as independent variables. A key element of creating an accurate and meaningful regression model is ensuring the variables are relevant and the model accounts for any interactions between them—more on this in the variable interaction section.

Data Analysis

Data analysis is the process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. The regression model mentioned in the exercise is a result of such data analysis, where historical data have been used to forecast future outcomes or identify trends. In practice, data analysis involves many steps: collecting data, preparing and cleaning it, choosing a suitable model, assessing model assumptions, and evaluating the model's predictive power.

When analyzing data, it's critical to understand the nuances of the model. For instance, including quadratic (\(x_2^2\)) or interaction terms (\(x_1x_2\)) can significantly improve the model's fit if such relationships are present in the data. Additionally, analyzing residual plots can help assess if there are patterns still unaccounted for, perhaps leading to further model refinement.

Variable Interaction

Variable interaction occurs when the effect of one independent variable on the dependent variable is different at different levels of another independent variable. In many real-world situations, variables are not independent of each other, and understanding their interaction is crucial for building an accurate model and correctly interpreting the results.

In the provided regression model, we see an interaction term represented by \(x_4 = x_1x_2\). This suggests that the impact of rotor speed (\(x_1\)) on tar content (\(y\)) varies depending on the gas inlet temperature (\(x_2\)) and vice versa. Because of this term, the effect of increasing \(x_2\) changes as \(x_1\) varies, which needs to be taken into account when interpreting the coefficient \(\beta_{2}\) as it does not represent a constant effect. This is why it is advised not to interpret \(\beta_{2}\) simply as the average change in tar content with a one-degree change in gas inlet temperature when rotor speed is constant, as this interpretation neglects the complex interaction depicted in the model.