4. Confidence and prediction intervals in multiple linear regression.

Recall that the least square estimate of the coefficients = (XX)-¹X¹Y follows

B~ N(B, o²(XX)-¹)

where N(μ, Σ) denotes the multivariate normal distribution with mean vector μ and

variance-covariance matrix Σ

(c) Find a 95% prediction interval for a new response at factor level 1 of A and factor

level 1 of B in problem 2 under a two-way ANOVA model without interaction. You may

use the value 10.025,13 = 2.16.

Note: The matrix XTX may be computed by hand. In R, you can create a matrix

by using the command

M<- matrix(c(1,2,3,4,5,6,7,8,9), nrow-3, ncol-3),

or simply

M <- matrix (1:9, 3, 3).

The inverse of the matrix can be found using the solve() function:

Minv<- solve (M).

Finally, if you have a vector v, then vMly may be computed by running

v %*% Minv * V.