PML

Nomor 1

x <- c(12,14,16,18,20,22)
y <- c(2.5,3,3.5,4.5,5,5.5)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  2.5
## [2,]  3.0
## [3,]  3.5
## [4,]  4.5
## [5,]  5.0
## [6,]  5.5
X
##         x
## [1,] 1 12
## [2,] 1 14
## [3,] 1 16
## [4,] 1 18
## [5,] 1 20
## [6,] 1 22
XtX <- t(X) %*% X
XtX
##          x
##     6  102
## x 102 1804
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##   [,1]
##     24
## x  430
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                        x
##    4.2952381 -0.24285714
## x -0.2428571  0.01428571
beta <- XtX_inv %*% Xty
beta
##         [,1]
##   -1.3428571
## x  0.3142857
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##     -1.3429       0.3143

Step 2

# Hitung Residual Sum Of Squaresnya

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 0.08571429
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 0.0003061224
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 0.01749636
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.776445
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.2657080 0.3628634

Soal Nomor 2

x <- c(100,150,200,250,300)
y <- c(3.2,4.1,5,5.8,6.5)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  3.2
## [2,]  4.1
## [3,]  5.0
## [4,]  5.8
## [5,]  6.5
X
##          x
## [1,] 1 100
## [2,] 1 150
## [3,] 1 200
## [4,] 1 250
## [5,] 1 300
XtX <- t(X) %*% X
XtX
##             x
##      5   1000
## x 1000 225000
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##     [,1]
##     24.6
## x 5335.0
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##               x
##    1.800 -8e-03
## x -0.008  4e-05
beta <- XtX_inv %*% Xty
beta
##     [,1]
##   1.6000
## x 0.0166
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##      1.6000       0.0166

Step 2

# Hitung Residual Sum Of Squaresnya

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 0.019
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 2.533333e-07
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 0.0005033223
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 3.182446
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.0149982 0.0182018

Nomor 3

x <- c(5,10,15,20,25,30)
y <- c(20,25,30,35,40,45)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]   20
## [2,]   25
## [3,]   30
## [4,]   35
## [5,]   40
## [6,]   45
X
##         x
## [1,] 1  5
## [2,] 1 10
## [3,] 1 15
## [4,] 1 20
## [5,] 1 25
## [6,] 1 30
XtX <- t(X) %*% X
XtX
##          x
##     6  105
## x 105 2275
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##   [,1]
##    195
## x 3850
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                         x
##    0.8666667 -0.040000000
## x -0.0400000  0.002285714
beta <- XtX_inv %*% Xty
beta
##   [,1]
##     15
## x    1
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##          15            1

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 1.729183e-27
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 9.881046e-31
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 9.940345e-16
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.776445
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 1 1
a<- t_alpha*se_beta1
a
## [1] 2.759882e-15

Nomor 4

x <- c(20,25,30,35,40,45,50)
y <- c(1.5,2,2.5,3,3.5,4,4.5)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  1.5
## [2,]  2.0
## [3,]  2.5
## [4,]  3.0
## [5,]  3.5
## [6,]  4.0
## [7,]  4.5
X
##         x
## [1,] 1 20
## [2,] 1 25
## [3,] 1 30
## [4,] 1 35
## [5,] 1 40
## [6,] 1 45
## [7,] 1 50
XtX <- t(X) %*% X
XtX
##          x
##     7  245
## x 245 9275
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##   [,1]
##     21
## x  805
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                        x
##    1.892857 -0.050000000
## x -0.050000  0.001428571
beta <- XtX_inv %*% Xty
beta
##   [,1]
##   -0.5
## x  0.1
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##        -0.5          0.1

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 1.203013e-28
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 3.43718e-32
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 1.853963e-16
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.570582
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.1 0.1

Nomor 5

x <- c(3,4,5,6,7,8)
y <- c(10,12,15,18,20,22)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]   10
## [2,]   12
## [3,]   15
## [4,]   18
## [5,]   20
## [6,]   22
X
##        x
## [1,] 1 3
## [2,] 1 4
## [3,] 1 5
## [4,] 1 6
## [5,] 1 7
## [6,] 1 8
XtX <- t(X) %*% X
XtX
##        x
##    6  33
## x 33 199
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##   [,1]
##     97
## x  577
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                        x
##    1.8952381 -0.31428571
## x -0.3142857  0.05714286
beta <- XtX_inv %*% Xty
beta
##       [,1]
##   2.495238
## x 2.485714
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##       2.495        2.486

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 0.7047619
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 0.01006803
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 0.1003396
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.776445
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 2.207127 2.764302

Nomor 6

x <- c(50,75,100,125,150,175)
y <- c(4.5,5,5.5,6,6.5,7)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  4.5
## [2,]  5.0
## [3,]  5.5
## [4,]  6.0
## [5,]  6.5
## [6,]  7.0
X
##          x
## [1,] 1  50
## [2,] 1  75
## [3,] 1 100
## [4,] 1 125
## [5,] 1 150
## [6,] 1 175
XtX <- t(X) %*% X
XtX
##           x
##     6   675
## x 675 86875
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##     [,1]
##     34.5
## x 4100.0
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                           x
##    1.32380952 -1.028571e-02
## x -0.01028571  9.142857e-05
beta <- XtX_inv %*% Xty
beta
##   [,1]
##   3.50
## x 0.02
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##        3.50         0.02

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 1.561945e-28
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 3.570159e-33
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 5.975081e-17
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.776445
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.02 0.02

Nomor 7

x <- c(200,250,300,350,400)
y <- c(150,175,200,225,250)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  150
## [2,]  175
## [3,]  200
## [4,]  225
## [5,]  250
X
##          x
## [1,] 1 200
## [2,] 1 250
## [3,] 1 300
## [4,] 1 350
## [5,] 1 400
XtX <- t(X) %*% X
XtX
##             x
##      5   1500
## x 1500 475000
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##     [,1]
##     1000
## x 312500
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                 x
##    3.800 -0.01200
## x -0.012  0.00004
beta <- XtX_inv %*% Xty
beta
##   [,1]
##   50.0
## x  0.5
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##        50.0          0.5

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 7.867909e-25
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 1.049055e-29
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 3.238911e-15
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 3.182446
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.5 0.5

Nomor 8

x <- c(5,10,15,20,25,30)
y <- c(60,70,75,85,90,95)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]   60
## [2,]   70
## [3,]   75
## [4,]   85
## [5,]   90
## [6,]   95
X
##         x
## [1,] 1  5
## [2,] 1 10
## [3,] 1 15
## [4,] 1 20
## [5,] 1 25
## [6,] 1 30
XtX <- t(X) %*% X
XtX
##          x
##     6  105
## x 105 2275
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##   [,1]
##    475
## x 8925
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                         x
##    0.8666667 -0.040000000
## x -0.0400000  0.002285714
beta <- XtX_inv %*% Xty
beta
##       [,1]
##   54.66667
## x  1.40000
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##       54.67         1.40

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 13.33333
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 0.007619048
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 0.08728716
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.776445
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 1.157652 1.642348

Nomor 9

x <- c(50,100,150,200,250)
y <- c(1.5,2,2.5,3,3.5)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  1.5
## [2,]  2.0
## [3,]  2.5
## [4,]  3.0
## [5,]  3.5
X
##          x
## [1,] 1  50
## [2,] 1 100
## [3,] 1 150
## [4,] 1 200
## [5,] 1 250
XtX <- t(X) %*% X
XtX
##            x
##     5    750
## x 750 137500
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##     [,1]
##     12.5
## x 2125.0
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##               x
##    1.100 -6e-03
## x -0.006  4e-05
beta <- XtX_inv %*% Xty
beta
##   [,1]
##   1.00
## x 0.01
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##        1.00         0.01

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 3.599178e-30
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 4.798904e-35
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 6.927412e-18
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 3.182446
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.01 0.01

Nomor 10

x <- c(100,150,200,250,300,350)
y <- c(2,2.5,3,3.5,4,4.5)
y <- matrix(y, ncol = 1)

X<- cbind(1,x)

y 
##      [,1]
## [1,]  2.0
## [2,]  2.5
## [3,]  3.0
## [4,]  3.5
## [5,]  4.0
## [6,]  4.5
X
##          x
## [1,] 1 100
## [2,] 1 150
## [3,] 1 200
## [4,] 1 250
## [5,] 1 300
## [6,] 1 350
XtX <- t(X) %*% X
XtX
##             x
##      6   1350
## x 1350 347500
##Cari X transpose kali y

Xty <- t(X) %*% y
Xty
##     [,1]
##     19.5
## x 4825.0
##Cari invers XtX

XtX_inv <- solve(XtX)
XtX_inv
##                            x
##    1.323809524 -5.142857e-03
## x -0.005142857  2.285714e-05
beta <- XtX_inv %*% Xty
beta
##   [,1]
##   1.00
## x 0.01
lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##        1.00         0.01

Step 2

y_pred <- X %*% beta

RSS <- sum((y - y_pred)^2)
RSS
## [1] 1.774937e-29
# Hitung Estimasi Varins sigma^2

n <- length(y)

sigma2 <- RSS/(n-2)

# Hitung Varian dan kesalahan standar dari Beta 1

var_beta1 <- sigma2 * XtX_inv[2,2]
var_beta1
## [1] 1.01425e-34
se_beta1 <- sqrt(var_beta1)
se_beta1
## [1] 1.0071e-17
#Hitung selang kepercayaan 95% untuk Beta 1 pakai uji t

alpha <- 0.05

t_alpha <- qt(1-alpha/2, n-2)
t_alpha
## [1] 2.776445
CI_beta1 <- c(beta[2] - t_alpha * se_beta1, beta[2] + t_alpha * se_beta1)
CI_beta1
## [1] 0.01 0.01