As this is a nested design the model equation is as follows:

\(y_{ijk}=\mu+\alpha_i+\beta_{j(i)}+\epsilon_{ijk}\)

Also the hypothesis is as follows:

\(H_0: \alpha_i=0\) and \(H_0:\sigma^2_\beta=0\)

\(H_1: \alpha_i\ne0\) and \(H_1:\sigma^2_\beta\ne0\)

process <- c(rep(1,4),rep(2,4),rep(3,4),rep(1,4),rep(2,4),rep(3,4),rep(1,4),rep(2,4),rep(3,4))
batch <- rep(seq(1,4),9)
obs <- c(25,19,15,15,19,23,18,35,14,35,38,25,30,28,17,16,17,24,21,27,15,21,54,29,26,20,14,13,14,21,17,25,20,24,50,33)
library(GAD)
process <- as.fixed(process)
batch <- as.random(batch)
data <- data.frame(process,batch,obs)
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  676.06  338.03  1.4643    0.2815    
## process:batch  9 2077.58  230.84 12.2031 5.477e-07 ***
## Residual      24  454.00   18.92                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(model)

summary(model)
## 
## Call:
## lm(formula = obs ~ process + batch %in% process)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -9.333 -2.083 -0.500  2.333  8.333 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       27.000      2.511  10.752 1.17e-10 ***
## process2         -10.333      3.551  -2.910  0.00768 ** 
## process3         -10.667      3.551  -3.004  0.00615 ** 
## process1:batch2   -4.667      3.551  -1.314  0.20123    
## process2:batch2    6.000      3.551   1.690  0.10406    
## process3:batch2   10.333      3.551   2.910  0.00768 ** 
## process1:batch3  -11.667      3.551  -3.285  0.00312 ** 
## process2:batch3    2.000      3.551   0.563  0.57853    
## process3:batch3   31.000      3.551   8.729 6.51e-09 ***
## process1:batch4  -12.333      3.551  -3.473  0.00197 ** 
## process2:batch4   12.333      3.551   3.473  0.00197 ** 
## process3:batch4   12.667      3.551   3.567  0.00156 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.349 on 24 degrees of freedom
## Multiple R-squared:  0.8585, Adjusted R-squared:  0.7936 
## F-statistic: 13.23 on 11 and 24 DF,  p-value: 1.177e-07

based on these results the effect of batches nested within the processes is significant (\(\alpha=5.477e-07\)), but the effect of the process was not significant (\(\alpha=0.05\)).