Question

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1. Consider the following model

y = β0 + β1x1 + β2x2 + β3x3 + ,

where y denotes the tool life and x1, x2, and x3 denote the cutting speed, tool type, and type of cutting oil, respectively. There are two different tool types, A and B, and there are two type of cutting oils, low-viscosity oil and medium-viscosity oil. The two categorical predictors are defined as x2 = (1 if type A 0 if type B) , and x3 = (1 if low-viscosity oil used 0 if medium-viscosity oil used)

(a) Interpret the regression coefficients β0, β1, β2 and β3. (Hint: The reference group is type B with medium-viscosity oil used - see slides 26-30 from Chapter 9 - Lecture 2.)

(b) Interpret the following hypothesis tests: (a) H0 : β2 = β3 = 0, and (b) H0 : β1 = 0.

Answer 1

a) Interpret the regression coefficients β0:

β0 is that the intercept of the regression curve that we try to suit . β0 is that the mean when all the variables X1, X2 & X3 are zero.

Interpret the regression coefficients β1:

is that the amount of change in "y" i.e. "tool life" for every unit increase within the X1 i.e. "cutting speed" keeping all the opposite variables constant i.e. for 1 unit increase in "cutting speed" the "tool life" will change by β1

Interpret the regression coefficients β2:

β2 that the difference between the mean of "tool type A" and therefore the mean of "tool type B". It are often interpreted because the "tool life" will change by β1 if the "tool type" may be a in comparison to B.

Interpret the regression coefficients β3:

β3 is that the difference between the mean of "if low-viscosity oil used" and therefore the mean of " if medium-viscosity oil used ". it's interpreted because the "tool life" will change by β3 if the "type of cutting oil" is "if low-viscosity oil used" in comparison to " if medium-viscosity oil used ".

b)

H0 : β2 = β3 = 0

If we build a rectilinear regression model with two variables x2 and x3 then we'll need to test if the variables are significant or not. We test it using the above hypothesis. If β2 , β3 are insignificant once we will have β2 = 0 & β3 = 0.

H0 : β1 = 0

If we build a rectilinear regression model with two variables x1 then we'll need to test if the variables are significant or not. We test it using the above hypothesis. If β1 are insignificant once we will have β1= 0

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