Show that, in a 23 design, Hint: Since both the right and
Chapter 9, Problem 2E(choose chapter or problem)
Show that, in a \(2^{3}\) design,
\(\begin{aligned}&\sum_{i=1}^{8}\left(X_{i}-\bar{X}\right)^{2}\\&\quad=8\left([\mathrm{~A}]^{2}+[\mathrm{B}]^{2}+[\mathrm{C}]^{2}+[\mathrm{AB}]^{2}+[\mathrm{AC}]^{2}+[\mathrm{BC}]^{2}+[\mathrm{ABC}]^{2}\right)\end{aligned}\)
HINT: Since both the right and the left members of this equation are symmetric in the variables \(X_{1}, X_{2}, \ldots, X_{8}\), it is necessary to show only that the corresponding coefficients of \(X_{1} X_{i}, i=1,2, \ldots, 8\), are the same in each member of the equation. Of course, recall that \(\bar{X}=\left(X_{1}+\right.\) \(\left.X_{2}+\cdots+X_{8}\right) / 8\).
Equation Transcription:
Text Transcription:
2^3
sum_i=1^8 ( X^i-bar X^2)^2 =8([A]^2+[B]^2+[C]^2+[AB]^2+[AC]^2+[BC]^2+[ABC]^2) X_1,X_2,…,X_8
X_1X_i, i=1,2,…,8
Bar X=(X_1+ X_2+⋯+X_8)/8
Unfortunately, we don't have that question answered yet. But you can get it answered in just 5 hours by Logging in or Becoming a subscriber.
Becoming a subscriber
Or look for another answer