1 Question 1: Is this a valid Latin Square? (explain)

Yes, it is a valid Latin Square since all letters appears in each row and each column.

2 Question 2: Write the model equation

Linear effect equation:

\[ Y_{ijk}=\mu+\tau_{i}+\beta_{j}+\alpha_{k}+\epsilon_{ijk} \]

Where,

µ = Grand Mean

τi = Treatment effect

βj = Block-1 effect

αk = Block-2 effect

εijk = Random error

i = number of treatments

j = number of block-1

k = number of block-2

3 Question 3: Analyze the data from this experiment (use α=0.05) and draw conclusions about the factor of interest.  (Note: Use aov() instead of gad() for Latin Square Designs, be sure all blocks are recognized as factors)

Null hypotheses = \(H_{0}:\mu_{ 1}=\mu_{ 2}=\mu_{ 3}=\mu_{ 4}=\mu_{ 5}\)

Alternative Hypotheses = At least one mu differs

Batch <- c(rep(1,5),rep(2,5),rep(3,5),rep(4,5),rep(5,5))
Day <- c(rep(seq(1,5),5))
Obs <- c(8,7,1,7,3,11,2,7,3,8,4,9,10,1,5,6,8,6,6,10,4,2,3,8,8)
Ingredients <- c("A","B","D","C","E","C","E","A","D","B","B","A","C","E","D","D","C","E","B","A","E","D","B","A","C")
Batch <- as.factor(Batch)
Day <- as.factor(Day)
Ingredients <- as.factor(Ingredients)
dat<- data.frame(Batch, Day, Obs, Ingredients)
model <- aov(Obs~Batch+Day+Ingredients)
summary(model)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## Batch        4  15.44    3.86   1.235 0.347618    
## Day          4  12.24    3.06   0.979 0.455014    
## Ingredients  4 141.44   35.36  11.309 0.000488 ***
## Residuals   12  37.52    3.13                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

P-value is larger than alpha 0.05, there is no evidence to reject Ho.

4 Complete R Code

Batch <- c(rep(1,5),rep(2,5),rep(3,5),rep(4,5),rep(5,5))
Day <- c(rep(seq(1,5),5))
Obs <- c(8,7,1,7,3,11,2,7,3,8,4,9,10,1,5,6,8,6,6,10,4,2,3,8,8)
Ingredients <- c("A","B","D","C","E","C","E","A","D","B","B","A","C","E","D","D","C","E","B","A","E","D","B","A","C")
Batch <- as.factor(Batch)
Day <- as.factor(Day)
Ingredients <- as.factor(Ingredients)
dat<- data.frame(Batch, Day, Obs, Ingredients)
model <- aov(Obs~Batch+Day+Ingredients)
summary(model)