1. Pendahuluan

Tujuan Praktikum:

1. Memahami konsep uji vektor nilai tengah untuk dua populasi.

2. Mampu melakukan uji hipotesis dan membentuk selang kepercayaan untuk:

Sampel Berpasangan (Paired Samples)
Sampel Saling Bebas (Independent Samples) dengan asumsi ragam sama.

3. Mampu menginterpretasikan hasil output R dalam konteks masalah.

2. Perbandingan Dua Vektor Nilai Tengah Sampel Berpasangan

Input Data

set.seed(63)

# Banyak baris data
n <- 15

# Bangkitkan data acak
data <- data.frame(
  `Mobil` = sample(100:230, n, replace = TRUE),
  `Mobil Matic` = sample(10:35, n, replace = TRUE),
  `Motor` = sample(70:220, n, replace = TRUE),
  `Motor Matic` = sample(7:30, n, replace = TRUE)
)

# Tampilkan data
print(data)

##    Mobil Mobil.Matic Motor Motor.Matic
## 1    166          17   153           7
## 2    141          29   134          21
## 3    104          20   155          28
## 4    184          31   184          17
## 5    112          14   155          20
## 6    190          35   159          19
## 7    151          20    74          25
## 8    183          20   112          29
## 9    101          11    86          30
## 10   223          12   165          20
## 11   197          23   198           7
## 12   105          11   186          19
## 13   208          24    93          27
## 14   134          25   128          19
## 15   191          30    90          26

Menghitung Selisih Essay Formal dan Informal

d1 = data[,1]-data[,3]
d2 = data[,2]-data[,4]

X = data.frame(d1,d2)
X

##     d1  d2
## 1   13  10
## 2    7   8
## 3  -51  -8
## 4    0  14
## 5  -43  -6
## 6   31  16
## 7   77  -5
## 8   71  -9
## 9   15 -19
## 10  58  -8
## 11  -1  16
## 12 -81  -8
## 13 115  -3
## 14   6   6
## 15 101   4

Menghitung vektor rataan dan matriks covarians

xbar = apply(X, 2, mean)
xbar

##         d1         d2 
## 21.2000000  0.5333333

cov_m = cov(X)
cov_m

##             d1         d2
## d1 3105.028571   9.457143
## d2    9.457143 115.980952

Uji T2 Hotelling Dua Populasi Sampel Berpasangan

library(MVTests)

## 
## Attaching package: 'MVTests'

## The following object is masked from 'package:datasets':
## 
##     iris

mean0 = c(0,0)
result = OneSampleHT2(X,mu0=mean0,alpha=0.05)
summary(result)

##        One Sample Hotelling T Square Test 
## 
## Hotelling T Sqaure Statistic = 2.199614 
##  F value = 1.021 , df1 = 2 , df2 = 13 , p-value: 0.387 
## 
##                    Descriptive Statistics
## 
##             d1         d2
## N     15.00000 15.0000000
## Means 21.20000  0.5333333
## Sd    55.72278 10.7694453
## 
## 
##                  Detection important variable(s)
## 
##         Lower     Upper Mu0 Important Variables?
## d1 -19.991167 62.391167   0                FALSE
## d2  -7.427612  8.494278   0                FALSE

Selang Kepercayaan Simultan

result$CI

##         Lower     Upper Mu0 Important Variables?
## d1 -19.991167 62.391167   0                FALSE
## d2  -7.427612  8.494278   0                FALSE

Selang Kepercayaan Bonferroni

bon = function(mu,S,n,alpha,k){
 p = length(mu)
 lower = mu[k] - sqrt(S[k,k]/n) * abs(qt(alpha/(2*p), df=n-1))
 upper = mu[k] + sqrt(S[k,k]/n) * abs(qt(alpha/(2*p), df=n-1))
 c(lower = lower,upper = upper)
}

n = nrow(X)

#Kata
bon(xbar, cov_m,n,0.05,1)

##  lower.d1  upper.d1 
## -14.90658  57.30658

#Kata Kerja
bon(xbar, cov_m,n,0.05,2)

##  lower.d2  upper.d2 
## -6.444923  7.511590

3. Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Sama

set.seed(63)
n <- 20

data2 <- data.frame(
  `UNBT SD`     = round(rnorm(n, mean = 65, sd = 10)), 
  `UNBT SMP`     = round(rnorm(n, mean = 55, sd = 12)),
  `UNBK SD`  = round(rnorm(n, mean = 70, sd = 15)),
  `UNBK SMP`  = round(rnorm(n, mean = 60, sd = 12))
)

print(data2)

##    UNBT.SD UNBT.SMP UNBK.SD UNBK.SMP
## 1       78       67      65       53
## 2       46       46      70       68
## 3       70       44      69       50
## 4       46       62      55       64
## 5       70       46      25       51
## 6       54       77      93       60
## 7       69       64      77       67
## 8       66       39      88       72
## 9       48       61      74       49
## 10      69       57      67       61
## 11      69       38      78       75
## 12      64       60      72       59
## 13      68       62      71       56
## 14      67       53      51       44
## 15      63       67      66       63
## 16      67       63      33       58
## 17      61       37      63       44
## 18      74       74      58       50
## 19      75       67      56       71
## 20      77       65      93       60

# Memisahkan Data dari Populasi 1 dan 2
d3 = data2[,1:2]
d4 = data2[,3:4]

Menghitung vektor rataan dan matriks covarians

xbar1 = apply(d3, 2, mean)
xbar1

##  UNBT.SD UNBT.SMP 
##    65.05    57.45

xbar2 = apply(d4, 2, mean)
xbar2

##  UNBK.SD UNBK.SMP 
##    66.20    58.75

cov_m1 = cov(d3)
cov_m1

##           UNBT.SD  UNBT.SMP
## UNBT.SD  92.78684  10.23947
## UNBT.SMP 10.23947 143.20789

cov_m2 = cov(d4)
cov_m2

##            UNBK.SD UNBK.SMP
## UNBK.SD  297.22105 56.36842
## UNBK.SMP  56.36842 83.25000

n1 = nrow(d3)
n2 = nrow(d4)

# S Gabungan
s_gab = ((n1-1)*cov_m1+(n2-1)*cov_m2)/(n1+n2-2)
s_gab

##            UNBT.SD  UNBT.SMP
## UNBT.SD  195.00395  33.30395
## UNBT.SMP  33.30395 113.22895

Uji T2 Hotelling Dua Populasi Sampel Saling Bebas Ragam Sama

library(Hotelling)

## Loading required package: corpcor

t2_homogen = hotelling.test(d3,d4,var.equal=TRUE)
t2_homogen

## Test stat:  0.18107 
## Numerator df:  2 
## Denominator df:  37 
## P-value:  0.9158

Selang Kepercayaan Simultan

T.ci = function(mu1, mu2, S_gab, n1, n2, avec=rep(1,length(mu)), level=0.95){
p = length(mu1)
mu = mu1-mu2
cval = qf(level, p, n1+n2-p-1) * p * (n1+n2-2) / (n1+n2-p-1)
zhat = crossprod(avec, mu)
zvar = crossprod(avec, S_gab %*% avec)* (1/n1+1/n2)
const = sqrt(cval * zvar)
c(lower = zhat - const, upper = zhat + const)
}

#TWK
T.ci(xbar1, xbar2, s_gab, n1,n2, avec=c(1,0),level=0.95)

##     lower     upper 
## -12.56295  10.26295

#TIU
T.ci(xbar1, xbar2, s_gab, n1,n2, avec=c(0,1),level=0.95)

##     lower     upper 
## -9.996707  7.396707

Selang Kepercayaan Bonferroni

bon = function(mu1, mu2 ,S, n1, n2, alpha, k){
 p = length(mu1)
 mu = mu1-mu2
 lower = mu[k] - sqrt((S[k,k]) *(1/n1+1/n2))* abs(qt(alpha/(2*p), df=n1+n2-2))
 upper = mu[k] + sqrt((S[k,k]) *(1/n1+1/n2))* abs(qt(alpha/(2*p), df=n1+n2-2))
 ci = c(lower = lower,upper = upper)
 names(ci)= c("lower","upper")
 ci
}

#TWK
bon(xbar1, xbar2, s_gab, n1, n2,0.05,1)

##      lower      upper 
## -11.455538   9.155538

#TIU
bon(xbar1, xbar2, s_gab, n1, n2,0.05,2)

##     lower     upper 
## -9.152857  6.552857

4. Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Tidak Sama

Input Data

set.seed(63)
n <- 20

data3 <- data.frame(
  `UNBT SD`     = round(rnorm(n, mean = 65, sd = 10)), 
  `UNBT SMP`     = round(rnorm(n, mean = 55, sd = 12)),
  `UNBK SD`  = round(rnorm(n, mean = 70, sd = 15)),
  `UNBK SMP`  = round(rnorm(n, mean = 60, sd = 12))
)

print(data3)

##    UNBT.SD UNBT.SMP UNBK.SD UNBK.SMP
## 1       78       67      65       53
## 2       46       46      70       68
## 3       70       44      69       50
## 4       46       62      55       64
## 5       70       46      25       51
## 6       54       77      93       60
## 7       69       64      77       67
## 8       66       39      88       72
## 9       48       61      74       49
## 10      69       57      67       61
## 11      69       38      78       75
## 12      64       60      72       59
## 13      68       62      71       56
## 14      67       53      51       44
## 15      63       67      66       63
## 16      67       63      33       58
## 17      61       37      63       44
## 18      74       74      58       50
## 19      75       67      56       71
## 20      77       65      93       60

# Memisahkan Data dari Populasi 1 dan 2
d5 = data3[,1:2]
d6 = data3[,3:4]

Menghitung vektor rataan dan matriks covarians

xbar1 = apply(d5, 2, mean)
xbar1

##  UNBT.SD UNBT.SMP 
##    65.05    57.45

xbar2 = apply(d5, 2, mean)
xbar2

##  UNBT.SD UNBT.SMP 
##    65.05    57.45

cov_m1 = cov(d5)
cov_m1

##           UNBT.SD  UNBT.SMP
## UNBT.SD  92.78684  10.23947
## UNBT.SMP 10.23947 143.20789

cov_m2 = cov(d6)
cov_m2

##            UNBK.SD UNBK.SMP
## UNBK.SD  297.22105 56.36842
## UNBK.SMP  56.36842 83.25000

n1 = nrow(d5)
n2 = nrow(d6)

Uji T2 Hotelling Dua Populasi Sampel Saling Bebas Ragam Sama

library(Hotelling)
t2_not_homogen = hotelling.test(d5,d6,var.equal=FALSE)
t2_not_homogen

## Test stat:  0.18107 
## Numerator df:  2 
## Denominator df:  33.9571171928145 
## P-value:  0.9158

Selang Kepercayaan Simultan

T.ci = function(mu1, mu2, S1, S2, n1, n2, avec=rep(1,length(mu)), level=0.95){
p = length(mu1)
mu = mu1-mu2
cval = qchisq(level, p)
zhat = crossprod(avec, mu)
zvar = crossprod(avec, S1 %*% avec)/n1 + crossprod(avec, S2 %*% avec)/n2
const = sqrt(cval * zvar)
c(lower = zhat - const, upper = zhat + const)
}

#TWK
T.ci(xbar1, xbar2, cov_m1, cov_m2, n1,n2, avec=c(1,0),level=0.95)

##     lower     upper 
## -10.80907  10.80907

#TWK
T.ci(xbar1, xbar2, cov_m1, cov_m2, n1,n2, avec=c(1,0),level=0.95)

##     lower     upper 
## -10.80907  10.80907

#TIU
T.ci(xbar1, xbar2, cov_m1, cov_m2, n1,n2, avec=c(0,1),level=0.95)

##     lower     upper 
## -8.236548  8.236548

Latihan

library(MASS)
set.seed(63)
unbt <- 20
unbk <- 20

# Nilai peserta CAT: Misalnya rata-rata lebih rendah
mu_unbt <- c(60, 55)
Sigma <- matrix(c(200, 50, 50, 180), 2) # Matriks kovariansi diasumsikan sama
data_unbt <- mvrnorm(unbt, mu = mu_unbt, Sigma = Sigma)

# Nilai peserta Manual: Misalnya rata-rata lebih tinggi
mu_unbk <- c(65, 60)
data_unbk <- mvrnorm(unbk, mu = mu_unbk, Sigma = Sigma)

# Gabungkan menjadi satu data frame
data_siswa <- data.frame(
  Metode = rep(c("UNBT", "UNBK"), each = 20),
  SD = c(data_unbt[,1], data_unbk[,1]),
  SMP = c(data_unbt[,2], data_unbk[,2])
)
data_siswa

##    Metode       SD      SMP
## 1    UNBT 51.54474 32.89156
## 2    UNBT 76.61587 80.59872
## 3    UNBT 47.53539 58.14733
## 4    UNBT 87.34108 68.17126
## 5    UNBT 47.94602 56.30959
## 6    UNBT 86.56771 48.83678
## 7    UNBT 60.90786 44.04124
## 8    UNBT 48.38442 65.92683
## 9    UNBT 83.47054 66.85448
## 10   UNBT 55.87114 49.65947
## 11   UNBT 45.12615 63.77619
## 12   UNBT 64.85345 52.60249
## 13   UNBT 60.15793 46.45980
## 14   UNBT 56.91586 54.38329
## 15   UNBT 70.30212 48.36083
## 16   UNBT 62.35042 46.80681
## 17   UNBT 53.18673 72.77664
## 18   UNBT 60.76063 31.71731
## 19   UNBT 55.87071 36.66395
## 20   UNBT 52.11118 35.45281
## 21   UNBK 64.27545 68.67255
## 22   UNBK 69.71600 54.23550
## 23   UNBK 59.17810 68.31901
## 24   UNBK 79.01712 66.87510
## 25   UNBK 95.84525 96.43944
## 26   UNBK 47.21205 44.89107
## 27   UNBK 63.78272 50.01694
## 28   UNBK 57.74188 39.09402
## 29   UNBK 55.46601 65.57480
## 30   UNBK 68.20733 60.82346
## 31   UNBK 67.41655 43.51128
## 32   UNBK 63.05253 59.77584
## 33   UNBK 61.64836 61.97146
## 34   UNBK 69.81975 84.47458
## 35   UNBK 70.36413 60.74942
## 36   UNBK 93.32125 86.34714
## 37   UNBK 60.93205 76.52587
## 38   UNBK 68.09878 75.43910
## 39   UNBK 83.17794 60.52428
## 40   UNBK 47.08078 44.75717

# Pisahkan data berdasarkan kelompok
group_unbt <- subset(data_siswa, Metode == "UNBT", select = c(SD, SMP))
group_unbt

##          SD      SMP
## 1  51.54474 32.89156
## 2  76.61587 80.59872
## 3  47.53539 58.14733
## 4  87.34108 68.17126
## 5  47.94602 56.30959
## 6  86.56771 48.83678
## 7  60.90786 44.04124
## 8  48.38442 65.92683
## 9  83.47054 66.85448
## 10 55.87114 49.65947
## 11 45.12615 63.77619
## 12 64.85345 52.60249
## 13 60.15793 46.45980
## 14 56.91586 54.38329
## 15 70.30212 48.36083
## 16 62.35042 46.80681
## 17 53.18673 72.77664
## 18 60.76063 31.71731
## 19 55.87071 36.66395
## 20 52.11118 35.45281

group_unbk <- subset(data_siswa, Metode == "UNBK", select = c(SD, SMP))
group_unbk

##          SD      SMP
## 21 64.27545 68.67255
## 22 69.71600 54.23550
## 23 59.17810 68.31901
## 24 79.01712 66.87510
## 25 95.84525 96.43944
## 26 47.21205 44.89107
## 27 63.78272 50.01694
## 28 57.74188 39.09402
## 29 55.46601 65.57480
## 30 68.20733 60.82346
## 31 67.41655 43.51128
## 32 63.05253 59.77584
## 33 61.64836 61.97146
## 34 69.81975 84.47458
## 35 70.36413 60.74942
## 36 93.32125 86.34714
## 37 60.93205 76.52587
## 38 68.09878 75.43910
## 39 83.17794 60.52428
## 40 47.08078 44.75717

P4 TPG Uji Vektor Rataan 2 Populasi

Tengku Muhammad Hafizh Fadhlah

2025-09-30

1. Pendahuluan

2. Perbandingan Dua Vektor Nilai Tengah Sampel Berpasangan

Input Data

Menghitung Selisih Essay Formal dan Informal

Menghitung vektor rataan dan matriks covarians

Uji T2 Hotelling Dua Populasi Sampel Berpasangan

Selang Kepercayaan Simultan

Selang Kepercayaan Bonferroni

3. Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Sama

Menghitung vektor rataan dan matriks covarians

Uji T2 Hotelling Dua Populasi Sampel Saling Bebas Ragam Sama

Selang Kepercayaan Simultan

Selang Kepercayaan Bonferroni

4. Perbandingan Dua Vektor Nilai Tengah Sampel Saling Bebas Ragam Tidak Sama

Input Data

Menghitung vektor rataan dan matriks covarians

Uji T2 Hotelling Dua Populasi Sampel Saling Bebas Ragam Sama

Selang Kepercayaan Simultan

Latihan