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2021-10-19

The article “Evaluating Vent Manifold Inerting Requirements: Flash Point Modeling for Organic Acid-Water Mixtures” presents a model to predict the flash point (in $$\displaystyle{F}^{{\circ}}$$) of a mixture of water, acetic acid, propionic acid, and butyric acid from the concentrations (in weight %) of the three acids. The results are as follows. The variable “Butyric Acid $$\displaystyle\times$$ Acetic Acid” is the interaction between butyric acid concentration and acetic acid concentration.
$$\begin{array}{|c|c|}\hline text{Predictor} & \text{Coef} & \text{SE Coef} & T & P \\ \hline \text{Constant} & 267.53 & 11.306 & 23.66 & 0.000 \\ \hline \text{Acetic Acid} & -1.5926 & 0.1295 & -12.30 & 0.000 \\ \hline \text{Propionic Acid} & -1.3897 & 0.1260 & -11.03 & 0.000 \\ \hline \text{Butyric Acid} & -1.0934 & 0.1164 & -9.39 & 0.000 \\ \hline \text{Butyric Acid} \times\text{Acetic Acid} & -0.002658 & 0.001145 & -2.32 & 0.034 \\ \hline \end{array}$$
a) Predict the flash point for a mixture that is 30% acetic acid, 35% propionic acid, and 30% butyric acid. (Note: In the model, 30% is represented by 30, not by 0.30.)
b) Someone asks by how much the predicted flash point will change if the concentration of acetic acid is increased by 10% while the other concentrations are kept constant. Is it possible to answer this question? If so, answer it. If not, explain why not.
c) Someone asks by how much the predicted flash point will change if the concentration of propionic acid is increased by 10% while the other concentrations are kept constant. Is it possible to answer this question? If so, answer it. If not, explain why not. Jaylen Fountain

Step 1
Let ${x}_{1}$ represent acetic acid, ${x}_{2}$ represent proportionic acid and ${x}_{3}$ represent butyric acid.
We note that the predictors are Constant, Acetic Acid, Propionic Acid, Butyric Acid and the product of butyric acid and acetic acid.
$y={\beta }_{0}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2}+{\beta }_{3}{x}_{3}+{\beta }_{4}{x}_{1}{x}_{3}$
The estimates of ${\beta }_{0},\cdots ,{\beta }_{4}$ are given in the column "SE Coef" of the Minitab output, which then results in the regression equation:
$\stackrel{^}{y}=267.53-1.5926{x}_{1}-1.3897{x}_{2}-1.0934{x}_{3}-0.002658{x}_{1}{x}_{3}$
Evaluate the regression equation at
${x}_{1}=30$
${x}_{2}=35$
${x}_{3}=35$
$\stackrel{^}{y}=267.53-1.5926\left(30\right)-1.3897\left(35\right)-1.0934\left(30\right)-0.002658\left(30\right)\left(30\right)$
Thus the predicted flash point is ${135.9183}^{\circ }F$
Step 2
Let ${x}_{1}$ represent acetic acid, ${x}_{2}$ represent proportionic acid and ${x}_{3}$ represent butyric acid.
We note that the predictors are Constant, Acetic Acid, Propionic Acid, Butyric Acid and the product of butyric acid and acetic acid.
$y={\beta }_{0}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2}+{\beta }_{3}{x}_{3}+{\beta }_{4}{x}_{1}{x}_{3}$
The acetic acid is represented bu ${x}_{1}$. Since ${x}_{1}$ occurs in the interaction term ${\beta }_{4}{x}_{1}{x}_{3}$ while the value of ${x}_{3}$ is unknown, we cannot determine the change in the predicted flash point $\stackrel{^}{y}$ when the value of ${x}_{1}$ changes.
Step 3
Let ${x}_{1}$ represent acetic acid, ${x}_{2}$ represent proportionic acid and ${x}_{3}$ represent butyric acid.
We note that the predictors are Constant, Acetic Acid, Propionic Acid, Butyric Acid and the product of butyric acid and acetic acid.
$y={\beta }_{0}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2}+{\beta }_{3}{x}_{3}+{\beta }_{4}{x}_{1}{x}_{3}$
We are intersted in the propionic acid, which is represented by the variable ${x}_{2}$. Since ${x}_{2}$ does not occur in any interaction termsm it is possible to determine the change in the predicted y-values.
The slope with respect to Propionic acid is given in the row "Propionic acid" and in the column "Coef" of the given MINITAB output:
$\stackrel{^}{{\beta }_{2}}=-1.3897$
When the x-values corresponding to Propionic acid differ by 10, then the predicted y-values will differ by 10 times the slope with respect to Propionic acid.
$10\stackrel{^}{{\beta }_{2}}=10\left(-1.3897\right)=-13.897$
Thus the predicted flashpoints will differ by ${13.897}^{\circ }F$

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