Question 1 a:

Design of factorial :

Factors = 3

Levels =2

we confound the three factor interaction abc with the blocks

Question 1 b:

Block 1: (1) ,ab,ac,bc

Block 2: a,b,c,abc

Question 2 a:

We confound the two factor interactions ab, ac and bc with the blocks.

Question 2 b:

Block 1: 1, abc
Block 2: a, bc
Block 3: b ac
Block 4: c ,ab

Half Normal Plot:

A<-c(-1,1,-1,1,-1,1,-1,1)
B<-c(-1,-1,1,1,-1,-1,1,1)
AB<-c(1,-1,-1,1,1,-1,-1,1)
C<-c(-1,-1,-1,-1,1,1,1,1)
AC<-c(1,-1,1,-1,-1,1,-1,1)
BC<-c(1,1,-1,-1,-1,-1,1,1)
ABC<-c(-1,1,1,-1,1,-1,-1,1)
obs<-c(22,32,35,55,44,40,60,39)
Data<-data.frame(A,B,AB,C,AC,BC,ABC)
Data
##    A  B AB  C AC BC ABC
## 1 -1 -1  1 -1  1  1  -1
## 2  1 -1 -1 -1 -1  1   1
## 3 -1  1 -1 -1  1 -1   1
## 4  1  1  1 -1 -1 -1  -1
## 5 -1 -1  1  1 -1 -1   1
## 6  1 -1 -1  1  1 -1  -1
## 7 -1  1 -1  1 -1  1  -1
## 8  1  1  1  1  1  1   1
library(DoE.base)
## Warning: package 'DoE.base' was built under R version 4.5.2
## Loading required package: grid
## Loading required package: conf.design
## Warning: package 'conf.design' was built under R version 4.5.2
## Registered S3 method overwritten by 'DoE.base':
##   method           from       
##   factorize.factor conf.design
## 
## Attaching package: 'DoE.base'
## The following objects are masked from 'package:stats':
## 
##     aov, lm
## The following object is masked from 'package:graphics':
## 
##     plot.design
## The following object is masked from 'package:base':
## 
##     lengths
?halfnormal
## starting httpd help server ...
##  done
Model <- lm(obs~A*B*C,data = Data)
coef(Model)
## (Intercept)           A           B           C         A:B         A:C 
##      40.875       0.625       6.375       4.875      -0.875      -6.875 
##         B:C       A:B:C 
##      -2.625      -3.375
halfnormal(Model)

## no significant effects

Question 3 a :

As per the above half-normal plot, none of the factors seem to be significant.

Question 3 b:

Also, as from the plot none of effects are significant thus, our blocks also don’t seem to be significant.