Part 1:
Yes, every letter only appears once in each column and row hence it is.
Part 2:
The linear equation would be:
\[y_{ijk} = \mu + \tau_i +\beta_j + \gamma_k + \epsilon_{ijk} \]
where \(\mu\) is the overall mean, \(\alpha_i\) is the factor of treatments, Both \(\beta_j\) and \(\gamma_k\) are blocks and \(\epsilon_{ijk}\) is the random error.
observation <- 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)
batch <- factor(rep(1:5, each = 5))
day <- c(rep(seq(1:5),5))
ingredient <- 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')
day <- as.factor(day)
ingredient <- as.factor(ingredient)
data <- data.frame(observation, batch, day, ingredient)
aov.model <- aov(observation~batch+day+ingredient, data=data)
summary(aov.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
## 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
The p-value of ingredient is 0.000488 is less than 0.05, so we reject the NULL hypothesis and accept that there is a significant effect of the ingredients in the reaction time.