Statistics for Environmental Engineers

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1


-1


1


-1


-1


1


-1


1


1


1


-1


+1


-1


-1


-1


-1


-1


1


1


-1


-1


1


1


-1


1


-1


1


-1


-1


-1


1


1


-1


-1


1


-1


1


1


1


1


1


1


1


X =


Notice that this matrix is the same as the model matrix for the 23 factorial shown in Table 27.3.


To calculate b and Var(b) we need the transpose of X, denoted as X’. The transpose is created by making the first column of X the first row of X’; the second column of X becomes the second row of X’, etc. This is shown below. We also need the product of X and X’, denoted as X’ X, the and the inverse of this, which is (X’ X)


The transpose of the X matrix is:


1


1


1


1


1


1


1


1


-1


1 —


1


1


-1


1


-1


1


-1


-1


1


1


-1


-1


1


1


X’ =


-1


-1 —


1


-1

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