1. interaction term

2.

pred22 <- function() {
   m <- rbind(c(1,4,5),c(1,2,1),c(2,3,4),c(2,1,1),c(3,4,2),c(3,1,5),c(4,4,1))
   mdf <- as.data.frame(m) 
   mdf$V1 <- as.factor(mdf$V1) 
   mdf$V2 <- as.factor(mdf$V2) 
   lmout <- lm(V3 ~ .,data=mdf) 
   newx <- mdf[3,]
   newx[,2] <- '2'
   predict(lmout,newx)
}

print(pred22())

3.  

cmpdepts <- function() 
{
   library(lme4)
   data(InstEval)
   ie <- InstEval
   tmp <- table(ie$dept,ie$y)
   f <- function(rw) rw %*% 1:5 / sum(rw)
   apply(tmp,1,f)
}

print(cmpdepts()[['8']])