#Question 1 The prompt or table given to us does represent a valid Latin Square. This is because the table is a proper 5x5. Another tell sign of it being a Latin Square is that the value is never repeated in the same row and or column. In other words, the values within the table appear once in every row and column.

#Question 2

Yij(k) i= reaction time with batch j= the day observed k= ingredient i= 1,…,5 j=1,…,5 u=mean ai=batch bj=day Eij(k)=random error Tk=ingredient/treatment

Yij(k)= u+ai+bj+Tk+Eij(k)

#Question3

batch<-factor(rep(1:5, each=5))
day<-factor(rep(1:5, times=5))
ingredient<-factor(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"))

y<-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)

data<-data.frame(batch,day,ingredient,y)

fit<-aov(y~batch+day+ingredient, data=data)
summary(fit)
##             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    
## ingredient   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
model.tables(fit, "means")
## Tables of means
## Grand mean
##      
## 5.88 
## 
##  batch 
## batch
##   1   2   3   4   5 
## 5.2 6.2 5.8 7.2 5.0 
## 
##  day 
## day
##   1   2   3   4   5 
## 6.6 5.6 5.4 5.0 6.8 
## 
##  ingredient 
## ingredient
##   A   B   C   D   E 
## 8.4 5.6 8.8 3.4 3.2
TukeyHSD(fit, "ingredient")
##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = y ~ batch + day + ingredient, data = data)
## 
## $ingredient
##     diff        lwr        upr     p adj
## B-A -2.8 -6.3646078  0.7646078 0.1539433
## C-A  0.4 -3.1646078  3.9646078 0.9960012
## D-A -5.0 -8.5646078 -1.4353922 0.0055862
## E-A -5.2 -8.7646078 -1.6353922 0.0041431
## C-B  3.2 -0.3646078  6.7646078 0.0864353
## D-B -2.2 -5.7646078  1.3646078 0.3365811
## E-B -2.4 -5.9646078  1.1646078 0.2631551
## D-C -5.4 -8.9646078 -1.8353922 0.0030822
## E-C -5.6 -9.1646078 -2.0353922 0.0023007
## E-D -0.2 -3.7646078  3.3646078 0.9997349

Based on wanting to analyze the data at a p or p-val of <.05, we can argue that there is a 95% family-wise confidence level. Moreover, we can say that the ingredients does and are the factor of interest. With ingredients C and A being the ones that have the most effect, based on their quantities (means) in all the batches observed.If you were wanting the quickest or faster reaction time, D and E are the ingredients to choose, since they have the lowest mean.