PML P5

Nomor 1

a

x1 <- matrix(c(4,3,6,7),,1)
y <- matrix(c(7,5,8,9),,1)
x <- cbind(1,x1)
x
##      [,1] [,2]
## [1,]    1    4
## [2,]    1    3
## [3,]    1    6
## [4,]    1    7
xt <- t(x)
xt
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    4    3    6    7
xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    4   20
## [2,]   20  110
xtx_i <- solve(xtx)
xtx_i
##       [,1] [,2]
## [1,]  2.75 -0.5
## [2,] -0.50  0.1
xty <- xt%*%y
xty
##      [,1]
## [1,]   29
## [2,]  154

b

b <- xtx_i %*% xty
b
##      [,1]
## [1,] 2.75
## [2,] 0.90

c

model <- lm(y~x1)
model
## 
## Call:
## lm(formula = y ~ x1)
## 
## Coefficients:
## (Intercept)           x1  
##        2.75         0.90
var_hat <- model$sigma^2

d

Ey1 <- b[1] + b[2]*x1[1]
Ey2 <- b[1] + b[2]*x1[2]
Ey3 <- b[1] + b[2]*x1[3]
Ey4 <- b[1] + b[2]*x1[4]
Ey <- t(cbind(Ey1,Ey2,Ey3,Ey4))
Ey
##     [,1]
## Ey1 6.35
## Ey2 5.45
## Ey3 8.15
## Ey4 9.05
var_hat1 <- model$sigma^2
var_hat1
## numeric(0)

Nomor 2

a

x2 <- matrix(c(2,3,4,5),,1)
y <- matrix(c(3,4,5,6),,1)
x <- cbind(1,x2)
x
##      [,1] [,2]
## [1,]    1    2
## [2,]    1    3
## [3,]    1    4
## [4,]    1    5
xt <- t(x)
xt
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    2    3    4    5
xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    4   14
## [2,]   14   54
xtx_i <- solve(xtx)
xtx_i
##      [,1] [,2]
## [1,]  2.7 -0.7
## [2,] -0.7  0.2
xty <- xt%*%y
xty
##      [,1]
## [1,]   18
## [2,]   68

b

b <- xtx_i %*% xty
b
##      [,1]
## [1,]    1
## [2,]    1

c

model <- lm(y~x2)
model
## 
## Call:
## lm(formula = y ~ x2)
## 
## Coefficients:
## (Intercept)           x2  
##           1            1
var_hat <- model$sigma^2
var_hat
## numeric(0)

d

Ey1 <- b[1] + b[2]*x2[1]
Ey2 <- b[1] + b[2]*x2[2]
Ey3 <- b[1] + b[2]*x2[3]
Ey4 <- b[1] + b[2]*x2[4]
Ey <- t(cbind(Ey1,Ey2,Ey3,Ey4))
Ey
##     [,1]
## Ey1    3
## Ey2    4
## Ey3    5
## Ey4    6
var_hat2 <- (sum((y-Ey)^2)) / (length(x2) - length(b))
var_hat2
## [1] 1.009742e-28

Nomor 3

a

x3 <- matrix(c(1,2,3,4),,1)
y <- matrix(c(10,12,14,16),,1)
x <- cbind(1,x3)
x
##      [,1] [,2]
## [1,]    1    1
## [2,]    1    2
## [3,]    1    3
## [4,]    1    4
xt <- t(x)
xt
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    1    2    3    4
xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    4   10
## [2,]   10   30
xtx_i <- solve(xtx)
xtx_i
##      [,1] [,2]
## [1,]  1.5 -0.5
## [2,] -0.5  0.2
xty <- xt%*%y
xty
##      [,1]
## [1,]   52
## [2,]  140

b

b <- xtx_i %*% xty
b
##      [,1]
## [1,]    8
## [2,]    2

c

model <- lm(y~x3)
model
## 
## Call:
## lm(formula = y ~ x3)
## 
## Coefficients:
## (Intercept)           x3  
##           8            2
var_hat <- model$sigma^2
var_hat
## numeric(0)

d

Ey1 <- b[1] + b[2]*x3[1]
Ey2 <- b[1] + b[2]*x3[2]
Ey3 <- b[1] + b[2]*x3[3]
Ey4 <- b[1] + b[2]*x3[4]
Ey <- t(cbind(Ey1,Ey2,Ey3,Ey4))
Ey
##     [,1]
## Ey1   10
## Ey2   12
## Ey3   14
## Ey4   16
var_hat3 <- (sum((y-Ey)^2)) / (length(x3) - length(b))
var_hat3
## [1] 0

Nomor 4

a

x4 <- matrix(c(1,2,4,6),,1)
y <- matrix(c(2,3,5,7),,1)
x <- cbind(1,x4)
x
##      [,1] [,2]
## [1,]    1    1
## [2,]    1    2
## [3,]    1    4
## [4,]    1    6
xt <- t(x)
xt
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    1    2    4    6
xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    4   13
## [2,]   13   57
xtx_i <- solve(xtx)
xtx_i
##            [,1]        [,2]
## [1,]  0.9661017 -0.22033898
## [2,] -0.2203390  0.06779661
xty <- xt%*%y
xty
##      [,1]
## [1,]   17
## [2,]   70

b

b <- xtx_i %*% xty
b
##      [,1]
## [1,]    1
## [2,]    1

c

model <- lm(y~x4)
model
## 
## Call:
## lm(formula = y ~ x4)
## 
## Coefficients:
## (Intercept)           x4  
##           1            1
var_hat <- model$sigma^2
var_hat
## numeric(0)

d

Ey1 <- b[1] + b[2]*x4[1]
Ey2 <- b[1] + b[2]*x4[2]
Ey3 <- b[1] + b[2]*x4[3]
Ey4 <- b[1] + b[2]*x4[4]
Ey <- t(cbind(Ey1,Ey2,Ey3,Ey4))
Ey
##     [,1]
## Ey1    2
## Ey2    3
## Ey3    5
## Ey4    7
var_hat4 <- (sum((y-Ey)^2)) / (length(x4) - length(b))
var_hat4
## [1] 1.676329e-30

Nomor 5

a

x5 <- matrix(c(5,3,6,8),,1)
y <- matrix(c(15,10,20,25),,1)
x <- cbind(1,x5)
x
##      [,1] [,2]
## [1,]    1    5
## [2,]    1    3
## [3,]    1    6
## [4,]    1    8
xt <- t(x)
xt
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    5    3    6    8
xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    4   22
## [2,]   22  134
xtx_i <- solve(xtx)
xtx_i
##            [,1]        [,2]
## [1,]  2.5769231 -0.42307692
## [2,] -0.4230769  0.07692308
xty <- xt%*%y
xty
##      [,1]
## [1,]   70
## [2,]  425

b

b <- xtx_i %*% xty
b
##           [,1]
## [1,] 0.5769231
## [2,] 3.0769231

c

model <- lm(y~x5)
model
## 
## Call:
## lm(formula = y ~ x5)
## 
## Coefficients:
## (Intercept)           x5  
##      0.5769       3.0769
var_hat <- model$sigma^2
var_hat
## numeric(0)

d

Ey1 <- b[1] + b[2]*x4[1]
Ey2 <- b[1] + b[2]*x4[2]
Ey3 <- b[1] + b[2]*x4[3]
Ey4 <- b[1] + b[2]*x4[4]
Ey <- t(cbind(Ey1,Ey2,Ey3,Ey4))
Ey
##          [,1]
## Ey1  3.653846
## Ey2  6.730769
## Ey3 12.884615
## Ey4 19.038462
var_hat5 <- (sum((y-Ey)^2)) / (length(x5) - length(b))
var_hat5
## [1] 112.7959

Nomor 6

a

x6 <- matrix(c(2000,2500,1800,2200,2400),,1)
y <- matrix(c(60,65,58,62,64),,1)
x <- cbind(1,x6)
x
##      [,1] [,2]
## [1,]    1 2000
## [2,]    1 2500
## [3,]    1 1800
## [4,]    1 2200
## [5,]    1 2400

b

xt <- t(x)

xtx <-xt%*%x
xtx
##       [,1]     [,2]
## [1,]     5    10900
## [2,] 10900 24090000
xtx_i <- solve(xtx)
xtx_i
##              [,1]          [,2]
## [1,] 14.689024390 -6.646341e-03
## [2,] -0.006646341  3.048780e-06
xty <- xt%*%y
xty
##        [,1]
## [1,]    309
## [2,] 676900

c

b1 <- ((length(x6)*sum(x6*y)) - (sum(x6)*sum(y))) / (length(x6)*sum(x6^2) - sum(x6)^2)
b1
## [1] 0.01

d

b0 <- (sum(x6^2)*sum(y) - sum(x6)*sum(x6*y)) / (length(x6)*sum(x6^2) - sum(x6)^2)

e <- y - b0 - (b1*x6)

s2 <- sum(e^2) / (length(x6) - 2)
s2
## [1] 0

e

var_b1 <- s2 / sum(x6^2)
var_b1
## [1] 0

Nomor 7

a

x7 <- matrix(c(1,2,3,4,5),,1)
y <- matrix(c(55,60,70,75,80),,1)
x <- cbind(1,x7)
x
##      [,1] [,2]
## [1,]    1    1
## [2,]    1    2
## [3,]    1    3
## [4,]    1    4
## [5,]    1    5

b

xt <- t(x)

xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    5   15
## [2,]   15   55
xtx_i <- solve(xtx)
xtx_i
##      [,1] [,2]
## [1,]  1.1 -0.3
## [2,] -0.3  0.1
xty <- xt%*%y
xty
##      [,1]
## [1,]  340
## [2,] 1085

c

b1 <- ((length(x7)*sum(x7*y)) - (sum(x7)*sum(y))) / (length(x7)*sum(x7^2) - sum(x7)^2)
b1
## [1] 6.5

d

b0 <- (sum(x7^2)*sum(y) - sum(x7)*sum(x7*y)) / (length(x7)*sum(x7^2) - sum(x7)^2)

e <- y - b0 - (b1*x7)

s2 <- sum(e^2) / (length(x7) - 2)
s2
## [1] 2.5

e

var_b1 <- s2 / sum(x7^2)
var_b1
## [1] 0.04545455

Nomor 8

a

x8 <- matrix(c(20,30,40,50,60),,1)
y <- matrix(c(60,65,58,62,64),,1)
x <- cbind(1,x8)
x
##      [,1] [,2]
## [1,]    1   20
## [2,]    1   30
## [3,]    1   40
## [4,]    1   50
## [5,]    1   60

b

xt <- t(x)

xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    5  200
## [2,]  200 9000
xtx_i <- solve(xtx)
xtx_i
##       [,1]   [,2]
## [1,]  1.80 -0.040
## [2,] -0.04  0.001
xty <- xt%*%y
xty
##       [,1]
## [1,]   309
## [2,] 12410

c

b1 <- ((length(x8)*sum(x8*y)) - (sum(x8)*sum(y))) / (length(x8)*sum(x8^2) - sum(x8)^2)
b1
## [1] 0.05

d

b0 <- (sum(x8^2)*sum(y) - sum(x8)*sum(x8*y)) / (length(x8)*sum(x8^2) - sum(x8)^2)

e <- y - b0 - (b1*x8)

s2 <- sum(e^2) / (length(x8) - 2)
s2
## [1] 10.1

e

var_b1 <- s2 / sum(x8^2)
var_b1
## [1] 0.001122222

Nomor 9

a

x9 <- matrix(c(100,200,300,400,500),,1)
y <- matrix(c(0.5,1,1.5,2,2.5),,1)
x <- cbind(1,x9)
x
##      [,1] [,2]
## [1,]    1  100
## [2,]    1  200
## [3,]    1  300
## [4,]    1  400
## [5,]    1  500

b

xt <- t(x)

xtx <-xt%*%x
xtx
##      [,1]   [,2]
## [1,]    5   1500
## [2,] 1500 550000
xtx_i <- solve(xtx)
xtx_i
##        [,1]   [,2]
## [1,]  1.100 -3e-03
## [2,] -0.003  1e-05
xty <- xt%*%y
xty
##        [,1]
## [1,]    7.5
## [2,] 2750.0

c

b1 <- ((length(x9)*sum(x9*y)) - (sum(x9)*sum(y))) / (length(x9)*sum(x9^2) - sum(x9)^2)
b1
## [1] 0.005

d

b0 <- (sum(x9^2)*sum(y) - sum(x9)*sum(x9*y)) / (length(x9)*sum(x9^2) - sum(x9)^2)

e <- y - b0 - (b1*x9)

s2 <- sum(e^2) / (length(x9) - 2)
s2
## [1] 0

e

var_b1 <- s2 / sum(x9^2)
var_b1
## [1] 0

Nomor 10

a

x10 <- matrix(c(1,2,3,4,5),,1)
y <- matrix(c(3,5,7,9,11),,1)
x <- cbind(1,x10)
x
##      [,1] [,2]
## [1,]    1    1
## [2,]    1    2
## [3,]    1    3
## [4,]    1    4
## [5,]    1    5

b

xt <- t(x)

xtx <-xt%*%x
xtx
##      [,1] [,2]
## [1,]    5   15
## [2,]   15   55
xtx_i <- solve(xtx)
xtx_i
##      [,1] [,2]
## [1,]  1.1 -0.3
## [2,] -0.3  0.1
xty <- xt%*%y
xty
##      [,1]
## [1,]   35
## [2,]  125

c

b1 <- ((length(x10)*sum(x10*y)) - (sum(x10)*sum(y))) / (length(x10)*sum(x10^2) - sum(x10)^2)
b1
## [1] 2

d

b0 <- (sum(x10^2)*sum(y) - sum(x10)*sum(x10*y)) / (length(x10)*sum(x10^2) - sum(x10)^2)

e <- y - b0 - (b1*x10)

s2 <- sum(e^2) / (length(x10) - 2)
s2
## [1] 0

e

var_b1 <- s2 / sum(10^2)
var_b1
## [1] 0