1 Question 1

We want to test the hypotheses of equal means against the hypotheses of at least one being different:

\[ H_0:\tau_i=0\\ H_a:\tau_i \neq 0 \] The linear effects model is shown by the equation below:

\[ y_{ij}=\mu_i +\tau_i + \beta_j +\epsilon_{ij}\] where \(\mu_i\), \(\tau_i\), \(\beta_j\), and \(\epsilon_{ij}\) represents, respectively the average of the ith treatment, the interaction between treatments, additive effect of the blocks and normal random error.

library(GAD)
chemical<-c(rep(1,5),rep(2,5),rep(3,5),rep(4,5))
bolt <- c(seq(1,5),seq(1,5),seq(1,5),seq(1,5))
obs<- c(73,68,74,71,67,
        73,67,75,72,70,
        75,68,78,73,68,
        73,71,75,75,69)

chemical<-as.fixed(chemical)
bolt <- as.fixed(bolt)
model<-lm(obs~chemical+bolt)
gad(model)
## Analysis of Variance Table
## 
## Response: obs
##          Df Sum Sq Mean Sq F value    Pr(>F)    
## chemical  3  12.95   4.317  2.3761    0.1211    
## bolt      4 157.00  39.250 21.6055 2.059e-05 ***
## Residual 12  21.80   1.817                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

With an \(\alpha=0.15\) the null hypotheses is rejected using the randomized block design with a p value of 0.1211.

2 Question 2

We want to test the hypotheses of equal means against the hypotheses of at least one being different:

\[ H_0:\tau_i=0\\ H_a:\tau_i \neq 0 \]

The linear effects model is shown by the equation below:

\[ y_{ij}=\mu_i +\tau_i +\epsilon_{ij}\]

chemical<-c(rep(1,5),rep(2,5),rep(3,5),rep(4,5))
obs<- c(73,68,74,71,67,
        73,67,75,72,70,
        75,68,78,73,68,
        73,71,75,75,69)

dat <- data.frame(obs,chemical)
dat$chemical <- as.factor(dat$chemical)
aov.model <- aov(obs~chemical,data=dat)
summary(aov.model)
##             Df Sum Sq Mean Sq F value Pr(>F)
## chemical     3  12.95   4.317   0.386  0.764
## Residuals   16 178.80  11.175
?aov

Using the ANOVA the null hypotheses is not rejected and the p-value was 0.764. The ANOVA does not account for the additive effect of the blocks. Moreover, the bolt type effect was neglected in this analysis, which might have influenced the answer.

3 Question 3

The bolt of cloth was probably the main reason for the not rejection of the null hypotheses. If we analyze the linear effects model from question 1 and 2, the only difference is the \(\beta\) term, i.e., the block effect. Therefore, we may assume that the completely randomized block design might be more accurate to perform an overral analysis of the entire process of fabrication.