A chemical engineer is investigating how the amount of conversion of a product from a raw material ( y ) depends on reaction temperature ( x 1 ) and the reaction time ( x 2 ). He has developed the...

A chemical engineer is investigating how the amount of conversion of a product from a raw material (y) depends on reaction temperature (x
₁) and the reaction time (x
₂). He has developed the following regression models:

1.
= 100 + 2x
₁+ 4x
₂

2.
 = 95 + 1.5x
₁+ 3x
₂+ 2x
₁
x
₂

Both models have been built over the range 0.5 ≤x
₂≤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
₁for 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
₁for 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?