In Exercise 7.7, if the true model is given by show that the expected values are given by E( 0 ) = 0

In Exercise 7.7, if the true model is given by

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In Exercise 7.7, if the true model is given by show that the expected values are given by E( 0 ) = 0
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show that the expected values are given by

E(0) = 0
123

E(b1), E(b2), E(b3) as in
Exercise 7:7

Exercises 7.7 and 7.8 illustrate the effect of aliasing in
the sense of a regression model. Thus, in the case of these two exercises, the
aliasing of xl with x2x3 implies that the
coefficient b1 is not an unbiased estimator of b1 but is biased by 23,
the coefficient of x2x3. Indeed, the coefficient of the
interaction described by the defining contrast, namely 123,
biases the estimated intercept 0.

Exercise 7.7

Consider the “model inadequacy” material in Section 7.2. The
material can be used to nicely illustrate the aliasing ideas discussed in
Chapter 4. Aliasing plays an important role when one deals with fractional
factorials for fitting first-order and first-order-plus-interaction response
surface models. Suppose one fits a first-order model using a 23-1III 
with defining relation

I = x1x2x3

However, suppose the true model is

Using Equation 7.2, we obtain

Use the expression

to verify the following expected values: