Q.1

We can write it for block design as ,

\(Y_{ij} = \mu + \tau_{i} + ß_{j}+ \epsilon_{ij}\)

Following is the hypothesis we are testing

Null Hypothesis : \(H_o : \tau_i = 0\)

Alternative Hypothesis : \(H_a : \tau_i \neq 0\) for some i

library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.
obs<-c(73,68,74,71,67,73,67,75,72,70,75,68,78,73,68,73,71,75,75,69)
bolt<-c(rep(seq(1,5),4))
chemical<-c(rep(1,5),rep(2,5),rep(3,5),rep(4,5))
bolt<-as.fixed(bolt)
chemical<-as.fixed(chemical)
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

As p value=0.1211 which is lesser than α= 0.15, we conclude that we reject the null hypothesis

Q.2

Without blocking the Linear effect equation is

Linear Effect Equation -> \(Y_{ij} = \mu_{ij} + \epsilon_{ij}\)

Following is the hypothesis we are testing

Null Hypothesis : \(H_o : \tau_i = 0\)

Alternative Hypothesis : \(H_a : \tau_i \neq 0\) for some i

boltt<-as.random(bolt)
boltt<-as.fixed(boltt)
model<-lm(obs~chemical)
gad(model)
## Analysis of Variance Table
## 
## Response: obs
##          Df Sum Sq Mean Sq F value Pr(>F)
## chemical  3  12.95  4.3167  0.3863 0.7644
## Residual 16 178.80 11.1750

As p value=0.7644 which is lesser than α= 0.15, we conclude that we fail to reject the null hypothesis

Q.3

From the results of Q.1 and Q.2 we can conclude that, Blocking allows us to account for sources of nuisance variability that are known as controllable.

DOF in CRD error is larger than RCBD

Hence, we do believe that the Bolt of cloth represents a significant amount of nuisance variability