# Problems 2 & 3 & 6

y <- c(14.0,32.0,11.6,21.2,17.4)
x <- c(16.0,21.7,6.0,16.2,11.0)

T <- length(y)

# Problem 3
b.hat <- (sum(x*y)-T*mean(x)*mean(y))/(sum(x^2)-T*mean(x)^2)
round(b.hat,1)

a.hat <- mean(y)-b.hat*mean(x) # Problem 2
round(a.hat,1)

# Using the lm function (linear model)

m <- lm(y~x)
summary(m)

# Using matrix algebra

ones <- rep(1,5)
ones

X <- cbind(ones,x)
X

params <- solve(t(X)%*%X)%*%t(X)%*%y
rownames(params) <- c("a.hat","b.hat")
colnames(params) <- c("Estimate")
round(params,1)

### Problem 6

A <- c(1.3,2.1,-1.4,2.1,0.8,1.9,-1.4,1.9,3.4)
A <- matrix(A,ncol=3)
A                      
B <- matrix(c(-1.6,2.9,0.8),ncol=1)
B
params <- A%*%B
rownames(params) <- c("beta1.hat","beta2.hat","beta3.hat")
colnames(params) <- c("Estimate")
params