1 Letter a

library(DoE.base)
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)
Observations<-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
Data<-data.frame(A,B,C,D,Observations)
model<-lm(Observations~A*B*C*D,data = Data)
coef(model)
##   (Intercept)             A             B             C             D 
##  1.737500e+01  2.250000e+00  2.500000e-01  1.000000e+00  1.625000e+00 
##           A:B           A:C           B:C           A:D           B:D 
## -3.750000e-01 -2.125000e+00  1.250000e-01  2.000000e+00 -8.543513e-17 
##           C:D         A:B:C         A:B:D         A:C:D         B:C:D 
##  1.110223e-16  5.000000e-01  3.750000e-01 -1.250000e-01 -3.750000e-01 
##       A:B:C:D 
##  5.000000e-01
halfnormal(model)

From the halfnormal plot, it appears that factors A and D are significant and the interaction between AC and AD are also significant.

2 Letter b

Now we want to test the hypotheses that the that the factors A, C, and the interactions are significant or not:

\[ H_0:\alpha_i=0\\ H_a:\alpha_i\neq0\\ H_0:\gamma_k=0\\ H_a:\gamma_k\neq0\\ H_0:\alpha\gamma_{ik}=0\\ H_a:\alpha\gamma_{ik}\neq0\\ H_0:\alpha\zeta_{iL}=0\\ H_a:\alpha\zeta_{iL}\neq0\\\]

model<-aov(Observations~A+C+A*C+A*D,data = Data)
summary(model)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1  81.00   81.00  49.846 3.46e-05 ***
## C            1  16.00   16.00   9.846 0.010549 *  
## D            1  42.25   42.25  26.000 0.000465 ***
## A:C          1  72.25   72.25  44.462 5.58e-05 ***
## A:D          1  64.00   64.00  39.385 9.19e-05 ***
## Residuals   10  16.25    1.62                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The results of the anova analysis show that the components A, C, D, AC, and AD are important at a significance level of 0.05.