2.

x1 <- c( rep( c(-1,1), times=9 ), rep(0,13) )
x2 <- c( rep( c(-1,1), each=2, times=4), rep(0,2), c(-1,1), rep(0,11))
x3 <- c( rep( c(-1,1), each=4, times=2), rep(0,4), c(-1,1), rep(0,9) )
x4 <- c( rep( c(-1,1), each=8, times=1), rep(0,6), c(-1,-1), rep(0,7) )
x5 <- c(1,-1,-1,1,-1,1,1,-1,-1,1,1,-1,1,-1,-1,1,rep(0,8),c(-1,1),rep(0,5) )
y <- c( 76.30,
   68.10,
   47.60,
   26.30,
   70.70,
   54.80,
   57.80,
   35.90,
   43.90,
   20.30,
   5.40,
   2.80,
   44.80,
   29.30,
   16.70,
   2.10,
   42.30,
   26.50,
   30.40,
   21.30,
   24.10,
   34.90,
   56.40,
   20.60,
   36.00,
   20.30,
   37.60,
   31.80,
   28.30,
   29.70,
   28.90
)

data <- data.frame(x1,x2,x3,x4,x5,y)

# (a)
cor(data)
#          x1    x2       x3       x4       x5       y
# x1 1.0000000 0.000000 0.00000000 0.0000000 0.00000000 -0.32165785
# x2 0.0000000 1.000000 0.00000000 0.0000000 0.00000000 -0.51386802
# x3 0.0000000 0.000000 1.00000000 0.0000000 0.00000000 0.07429973
# x4 0.0000000 0.000000 0.00000000 1.0000000 0.00000000 -0.64851745
# x5 0.0000000 0.000000 0.00000000 0.0000000 1.00000000 -0.09898939
# y -0.3216578 -0.513868 0.07429973 -0.6485175 -0.09898939 1.00000000
#
# There are no variables that are really correlated with one another. A pair of variables
# that are the most highly correlated is x4 and y with the correlation of merely -0.64.

# (b)
fit <- lm(y~x1+x2+x3+x4+x5, data=data)
predict(fit, newdata=data.frame(x1=37.5, x2=2.2, x3=10, x4=5, x5=100),
   interval="prediction", level=0.95 )
#       fit    lwr    upr
# 1 -582.8656 -1055.54 -110.1913

# (c)
fit <- lm(y~x1+x2

PROBLEM 2 Suppose Y follows a bivariate normal distribution with p...