a.Display the half-normal plot for this data and determine which factors appear to be significant.

library(DoE.base)
## Warning: package 'DoE.base' was built under R version 4.2.2
## Loading required package: grid
## Loading required package: conf.design
## 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
A<-c(-1,+1,-1,+1,-1,+1,-1,+1,-1,+1,-1,+1,-1,+1,-1,+1)
B<-c(-1,-1,+1,+1,-1,-1,+1,+1,-1,-1,+1,+1,-1,-1,+1,+1)
C<-c(-1,-1,-1,-1,+1,+1,+1,+1,-1,-1,-1,-1,+1,+1,+1,+1)
D<-c(-1,-1,-1,-1,-1,-1,-1,-1,+1,+1,+1,+1,+1,+1,+1,+1)
obs<-c(12,18,13,20,17,25,15,25,10,24,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,obs)
model<-aov(obs~A*B*C*D,data = dat)
halfnormal(model)
## 
## Significant effects (alpha=0.05, Lenth method):
## [1] A     C     A:C:D

the factors A,C,D are the factors that are significant

b.Pull terms that do not appear to be significant into error and test for the significance of the other effects at the 0.05 level of significance. State the hypothesis to be tested, perform your analysis, and then conclude.

model1<-aov(obs~A+C+D+A*C*D,data = dat)
summary(model1)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1 256.00  256.00 157.538 1.52e-06 ***
## C            1  49.00   49.00  30.154  0.00058 ***
## D            1   2.25    2.25   1.385  0.27314    
## A:C          1   9.00    9.00   5.538  0.04643 *  
## A:D          1   0.25    0.25   0.154  0.70513    
## C:D          1   6.25    6.25   3.846  0.08550 .  
## A:C:D        1  30.25   30.25  18.615  0.00256 ** 
## Residuals    8  13.00    1.63                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

all the factors A,C,D are seemed to be Significant from the halfnormal plot with the significance of 0.05