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## Exercises

1. Suppose we are analyzing a set of 4 samples. The first two samples are from a treatment group A and the second two samples are from a treatment group B. This design can be represented with a model matrix like so:

 X <- matrix(c(1,1,1,1,0,0,1,1),nrow=4)
rownames(X) <- c("a","a","b","b")
X

 ##   [,1] [,2]
## a    1    0
## a    1    0
## b    1    1
## b    1    1


Suppose that the fitted parameters for a linear model give us:

 beta <- c(5, 2)


Use the matrix multiplication operator, %*%, in R to answer the following questions:

What is the fitted value for the A samples? (The fitted Y values.)

2. What is the fitted value for the B samples? (The fitted Y values.)

3. Suppose now we are comparing two treatments B and C to a control group A, each with two samples. This design can be represented with a model matrix like so:

 X <- matrix(c(1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,1),nrow=6)
rownames(X) <- c("a","a","b","b","c","c")
X


Suppose that the fitted values for the linear model are given by:

 beta <- c(10,3,-3)


What is the fitted value for the B samples?

4. What is the fitted value for the C samples?