PART A:
Model Equation:
y_ijk = mu + a_i + b_j(i) + e_ijk where i = 1,2,3 j= 1,2,3,4 and k= 1,2,3
PART B:
Null : a_i = 0
ALternative: a_i \(\neq\) 0
Null: b_j(i)=0
Alternative: b_j(i) \(\neq\) 0
PART C:
library(GAD)
process<-c(rep(1,12),rep(2,12),rep(3,12))
batch<-rep(c(rep(1,3),rep(2,3),rep(3,3),rep(4,3)),3)
obs<-c(25,30,26,19,28,20,15,17,14,15,16,13,29,27,24,23,24,21,28,21,27,35,27,25,24,25,20,35,21,24,38,34,30,25,29,33)
Table<-data.frame(process,batch,obs)
process<-as.fixed(process)
batch<-as.random(batch)
model<-lm(obs~process+batch%in%process)
gad(model)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## process 2 446.06 223.028 3.5365 0.073563 .
## process:batch 9 567.58 63.065 4.1965 0.002349 **
## Residual 24 360.67 15.028
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1--> The process is not sifnificant(so we fail to reject the null hypothesis ) because it has a P-value (0.0735) which is greater than alpha=0.05
But Batch nested with process is significant (successfully reject the null hypothesis) because it has a P-value (0.0023) which is smaller than alpha = 0.05