(a) Model Equation:

Y_ijk=mu+alpha_i+beta_j+beta_j(i)+E_ijk

  1. Hypothesis:

Null hypothesis; Ho: Beta_j(i)=0

Alternative Hypothesis Ha; Beta_j(i)=/0

  1. Analysis:
library(GAD)
## Warning: package 'GAD' was built under R version 4.1.3
## Loading required package: matrixStats
## Warning: package 'matrixStats' was built under R version 4.1.3
## Loading required package: R.methodsS3
## Warning: package 'R.methodsS3' was built under R version 4.1.3
## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.
obs<-c(25,19,15,15,29,23,28,35,24,35,38,25,30,28,17,16,27,24,21,27,25,21,34,29,26,20,14,13,24,21,27,25,20,24,30,33)
process<-c(1,1,1,1,2,2,2,2,3,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3)
batch<-rep(rep(c(1,2,3,4),3),3)
process<-as.fixed(process)
batch<-as.random(batch)
model<-aov(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

Comments: So we have sufficient evidence to reject the null hypothesis Ho, that the nested within the factor is not effecting the model. We took process has primary factor and batch as random factor by looking P-value from GAD model, we got the factor within factor as significant at 0.05.