Question
A chemical engineer is investigating how the amount of conversion of a product from a raw material $(y)$ depends on reaction temperature $\left(x_{1}\right)$ and the reaction time $\left(x_{2}\right) .$ He has developed the following regression models:$$ \begin{array}{l} \text { 1. } \hat{y}=100+2 x_{1}+4 x_{2} \\ \text { 2. } \hat{y}=95+1.5 x_{1}+3 x_{2}+2 x_{1} x_{2} \end{array} $$Both models have been built over the range $0.5 \leq x_{2} \leq 10$.(a) What is the predicted value of conversion when $x_{2}=2 ?$ Repeat this calculation for $x_{2}=8 .$ Draw a graph of the predicted values for both conversion models. Comment on the effect of the interaction term in model 2 .(b) Find the expected change in the mean conversion for a unit change in temperature $x_{1}$ for model 1 when $x_{2}=5 .$ Does this quantity depend on the specific value of reaction time selected? Why?(c) Find the expected change in the mean conversion for a unit change in temperature $x_{1}$ for model 2 when $x_{2}=5 .$ Repeat this calculation for $x_{2}=2$ and $x_{2}=8$. Does the result depend on the value selected for $x_{2}$ ? Why?
Step 1
Using model 1: \[ \hat{y} = 100 + 2x_{1} + 4x_{2} \] Substituting \( x_{2} = 2 \): \[ \hat{y} = 100 + 2x_{1} + 4(2) = 100 + 2x_{1} + 8 = 108 + 2x_{1} \] Show more…
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