Problem 13

Entering and sorting the data in r:

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.
obs13 <- 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)
batch <- rep(rep(c(1,2,3,4),3),3)
process <- rep(c(rep(1,4), rep(2,4), rep(3,4)),3)  

batch <- as.random(batch)
process <- as.fixed(process)

Problem 13(a)

model eqn:

yijk= mu+ai +bj(i) + eijk

Where, mu = grand mean

ai = higher level factor (process)

bj(i) = lower level factor (batch nested in process)

eijk = random error

Problem 13(b)

Null Hypothesis, Ho: ai = 0

bj(i) = 0

Alternative hypothesis, Ha: ai ≠ 0

bj(i) ≠ 0

Problem 13(c)

model13 <- aov(obs13~process+batch%in%process)
GAD::gad(model13)
## Analysis of Variance Table
## 
## Response: obs13
##               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

From our ANOVA analysis, we see that our lower level random factor (batch nested in process) show significant main effect with p-value (0.002) < level of significance of alpha = 0.05. The higher level fixed factor (process) does not show significant effect with p-value higher than the significance level.