Write out a 22 design, displaying the A, B, and AB columns

Chapter 9, Problem 1E

(choose chapter or problem)

Write out a 22 design, displaying the A, B, and AB columns for the four runs.

(a) If X1,X2,X3, and X4 are the four observations for the respective runs in standard order, write out the three linear forms, [A], [B], and [AB], that measure the two main effects and the interaction. These linear forms should include the divisor 22 = 4.

(b) Show that (Xi - )2 = 4([A]2 + [B]2 + [AB]2).

(c) Under the null hypothesis that all the means are equal and with the usual assumptions (normality, mutual independence, and common variance), what can you say about the distributions of the expressions in (b) after each is divided by σ2?

Write out a \(2^{2}\) design, displaying the \(\mathrm{A}, \mathrm{B}\), and $\mathrm{AB}$ columns for the four runs.

(a) If \(X_{1}, X_{2}, X_{3}$, and $X_{4}$ are the four observations for the respective runs in standard order, write out the three linear forms, $[\mathrm{A}],[\mathrm{B}]$, and $[\mathrm{AB}]$, that measure the two main effects and the interaction. These linear forms should include the divisor $2^{2}=4$.

(b) Show that $\sum_{i=1}^{4}\left(X_{i}-\bar{X}\right)^{2}=4\left([\mathrm{~A}]^{2}+[\mathrm{B}]^{2}+[\mathrm{AB}]^{2}\right)$.

(c) Under the null hypothesis that all the means are equal and with the usual assumptions (normality, mutual independence, and common variance), what can you say about the distributions of the expressions in (b) after each is divided by $\sigma^{2}$ ?

Equation Transcription:

 

Text Transcription:

A,B

AB  

X_1,X_2,X_3,  

X_4  

[A],[B]

[AB]

2^2=4

sum_i=1^4 (X_i-bar X)^2=4([ A]^2+[B]^2+[AB]^2)

sigma^2