Estimasi Distribusi dan Parameter Model

Teorema Limit Pusat

Rata-rata dari contoh acak yang berasal dari sebaran apapun memiliki sebaran normal jika ukuran contohnya sangat besar
Jika contoh acak diambil dari populasi dengan mean $\mu$ dan ragam $\sigma^{2}$ , maka semakin besar ukuran contoh, sebaran dari $\bar{x}$ akan semakin mendekati sebaran normal dengan mean $\mu$ dan ragam $\frac{σ^2}n$

Algoritma

Tentukan ukuran contoh ( $n$ )
Tentukan sebaran data
Ulang $k$ kali

Ambil $n$ contoh acak dari sebaran data yang sudah ditentukan
Hitung rataannya lalu simpan

Periksa sebaran dari $𝑘$ rataan

Aplikasi di R

Hampiran Normal Terhadap Geometrik

par(mfrow=c(3,1))
library(probs)

## Warning: package 'probs' was built under R version 4.4.3

## 
## Attaching package: 'probs'

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, union

set.seed(123)
populasi    = rgeom(20, 0.1)

n1          = 2
contoh_geo1 = urnsamples(populasi, size = 2, replace = F, ordered = F)
mean_geo1   = matrix(apply(contoh_geo1, 1, mean))

n2          = 5
contoh_geo2 = urnsamples(populasi, size = 5, replace = F, ordered = F)
mean_geo2   = matrix(apply(contoh_geo2, 1, mean))

n3          = 10
contoh_geo3 = urnsamples(populasi, size = 10, replace = F, ordered = F)
mean_geo3   = matrix(apply(contoh_geo3, 1, mean))

hist(mean_geo1,main = paste("Hampiran Normal Terhadap Geometrik (n = 2)"),xlab = "xbar")
hist(mean_geo2,main = paste("Hampiran Normal Terhadap Geometrik (n = 5)"),xlab = "xbar")
hist(mean_geo3,main = paste("Hampiran Normal Terhadap Geometrik (n = 10)"),xlab = "xbar")

Tujuannya adalah untuk menunjukkan bagaimana distribusi rata-rata sampel $\bar{x}$ dari distribusi geometrik mendekati distribusi normal ketika ukuran sampel $(n)$ meningkat.

Hampiran Normal Terhadap Eksponensial

par(mfrow=c(3,1))
library(probs)
set.seed(123)
populasi    = rexp(20)

n1          = 2
contoh_exp1 = urnsamples(populasi, size = 2, replace = F, ordered = F)
mean_exp1   = matrix(apply(contoh_exp1, 1, mean))

n2          = 5
contoh_exp2 = urnsamples(populasi, size = 5, replace = F, ordered = F)
mean_exp2   = matrix(apply(contoh_exp2, 1, mean))

n3          = 10
contoh_exp3 = urnsamples(populasi, size = 10, replace = F, ordered = F)
mean_exp3   = matrix(apply(contoh_exp3, 1, mean))

hist(mean_exp1,main = paste("Hampiran Normal Terhadap Eksponensial (n = 2)"),xlab = "xbar")
hist(mean_exp2,main = paste("Hampiran Normal Terhadap Eksponensial (n = 5)"),xlab = "xbar")
hist(mean_exp3,main = paste("Hampiran Normal Terhadap Eksponensial (n = 10)"),xlab = "xbar")

Hampiran Normal Terhadap Seragam

par(mfrow=c(3,1))
library(probs)
set.seed(123)
populasi     = runif(20)

n1           = 2
contoh_unif1 = urnsamples(populasi, size = 2, replace = F, ordered = F)
mean_unif1   = matrix(apply(contoh_unif1, 1, mean))

n2           = 5
contoh_unif2 = urnsamples(populasi, size = 5, replace = F, ordered = F)
mean_unif2   = matrix(apply(contoh_unif2, 1, mean))

n3           = 10
contoh_unif3 = urnsamples(populasi, size = 10, replace = F, ordered = F)
mean_unif3   = matrix(apply(contoh_unif3, 1, mean))

hist(mean_unif1,main = paste("Hampiran Normal Terhadap Seragam (n = 2)"),xlab = "xbar")
hist(mean_unif2,main = paste("Hampiran Normal Terhadap Seragam (n = 5)"),xlab = "xbar")
hist(mean_unif3,main = paste("Hampiran Normal Terhadap Seragam (n = 10)"),xlab = "xbar")

Kesimpulan : Semakin besar ukuran contoh, maka sebaran rata-rata dari contoh acak yang berasal dari sebaran geometrik, eksponensial, maupun uniform akan mendekati sebaran normal. Hal ini ditunjukkan dari histogram yang mana ketika n semakin besar akan semakin cenderung membentuk kurva normal

Sebaran Percontohan Sebaran Normal

par(mfrow=c(3,1))
library(probs)
set.seed(1299)
populasi     = rnorm(20,5,sqrt(12)) # Membangkitkan bil. acak ~ Normal (miu = 5, sigma2 =12) 

n1           = 3
contoh_norm1 = urnsamples(populasi, size = 3, replace = F, ordered = F)
contoh_norm1

##              X1         X2         X3
## 1     1.2759246  4.9063860  0.9127462
## 2     1.2759246  4.9063860  4.7962633
## 3     1.2759246  4.9063860 -0.4960505
## 4     1.2759246  4.9063860  1.3199713
## 5     1.2759246  4.9063860  9.1003173
## 6     1.2759246  4.9063860  7.8230413
## 7     1.2759246  4.9063860  3.9885906
## 8     1.2759246  4.9063860 -1.2761191
## 9     1.2759246  4.9063860  7.1127273
## 10    1.2759246  4.9063860  1.0256657
## 11    1.2759246  4.9063860 15.8759113
## 12    1.2759246  4.9063860  6.6170191
## 13    1.2759246  4.9063860  5.9357163
## 14    1.2759246  4.9063860  7.1021421
## 15    1.2759246  4.9063860  8.1282322
## 16    1.2759246  4.9063860  6.0001488
## 17    1.2759246  4.9063860  3.6230786
## 18    1.2759246  4.9063860  2.4165908
## 19    1.2759246  0.9127462  4.7962633
## 20    1.2759246  0.9127462 -0.4960505
## 21    1.2759246  0.9127462  1.3199713
## 22    1.2759246  0.9127462  9.1003173
## 23    1.2759246  0.9127462  7.8230413
## 24    1.2759246  0.9127462  3.9885906
## 25    1.2759246  0.9127462 -1.2761191
## 26    1.2759246  0.9127462  7.1127273
## 27    1.2759246  0.9127462  1.0256657
## 28    1.2759246  0.9127462 15.8759113
## 29    1.2759246  0.9127462  6.6170191
## 30    1.2759246  0.9127462  5.9357163
## 31    1.2759246  0.9127462  7.1021421
## 32    1.2759246  0.9127462  8.1282322
## 33    1.2759246  0.9127462  6.0001488
## 34    1.2759246  0.9127462  3.6230786
## 35    1.2759246  0.9127462  2.4165908
## 36    1.2759246  4.7962633 -0.4960505
## 37    1.2759246  4.7962633  1.3199713
## 38    1.2759246  4.7962633  9.1003173
## 39    1.2759246  4.7962633  7.8230413
## 40    1.2759246  4.7962633  3.9885906
## 41    1.2759246  4.7962633 -1.2761191
## 42    1.2759246  4.7962633  7.1127273
## 43    1.2759246  4.7962633  1.0256657
## 44    1.2759246  4.7962633 15.8759113
## 45    1.2759246  4.7962633  6.6170191
## 46    1.2759246  4.7962633  5.9357163
## 47    1.2759246  4.7962633  7.1021421
## 48    1.2759246  4.7962633  8.1282322
## 49    1.2759246  4.7962633  6.0001488
## 50    1.2759246  4.7962633  3.6230786
## 51    1.2759246  4.7962633  2.4165908
## 52    1.2759246 -0.4960505  1.3199713
## 53    1.2759246 -0.4960505  9.1003173
## 54    1.2759246 -0.4960505  7.8230413
## 55    1.2759246 -0.4960505  3.9885906
## 56    1.2759246 -0.4960505 -1.2761191
## 57    1.2759246 -0.4960505  7.1127273
## 58    1.2759246 -0.4960505  1.0256657
## 59    1.2759246 -0.4960505 15.8759113
## 60    1.2759246 -0.4960505  6.6170191
## 61    1.2759246 -0.4960505  5.9357163
## 62    1.2759246 -0.4960505  7.1021421
## 63    1.2759246 -0.4960505  8.1282322
## 64    1.2759246 -0.4960505  6.0001488
## 65    1.2759246 -0.4960505  3.6230786
## 66    1.2759246 -0.4960505  2.4165908
## 67    1.2759246  1.3199713  9.1003173
## 68    1.2759246  1.3199713  7.8230413
## 69    1.2759246  1.3199713  3.9885906
## 70    1.2759246  1.3199713 -1.2761191
## 71    1.2759246  1.3199713  7.1127273
## 72    1.2759246  1.3199713  1.0256657
## 73    1.2759246  1.3199713 15.8759113
## 74    1.2759246  1.3199713  6.6170191
## 75    1.2759246  1.3199713  5.9357163
## 76    1.2759246  1.3199713  7.1021421
## 77    1.2759246  1.3199713  8.1282322
## 78    1.2759246  1.3199713  6.0001488
## 79    1.2759246  1.3199713  3.6230786
## 80    1.2759246  1.3199713  2.4165908
## 81    1.2759246  9.1003173  7.8230413
## 82    1.2759246  9.1003173  3.9885906
## 83    1.2759246  9.1003173 -1.2761191
## 84    1.2759246  9.1003173  7.1127273
## 85    1.2759246  9.1003173  1.0256657
## 86    1.2759246  9.1003173 15.8759113
## 87    1.2759246  9.1003173  6.6170191
## 88    1.2759246  9.1003173  5.9357163
## 89    1.2759246  9.1003173  7.1021421
## 90    1.2759246  9.1003173  8.1282322
## 91    1.2759246  9.1003173  6.0001488
## 92    1.2759246  9.1003173  3.6230786
## 93    1.2759246  9.1003173  2.4165908
## 94    1.2759246  7.8230413  3.9885906
## 95    1.2759246  7.8230413 -1.2761191
## 96    1.2759246  7.8230413  7.1127273
## 97    1.2759246  7.8230413  1.0256657
## 98    1.2759246  7.8230413 15.8759113
## 99    1.2759246  7.8230413  6.6170191
## 100   1.2759246  7.8230413  5.9357163
## 101   1.2759246  7.8230413  7.1021421
## 102   1.2759246  7.8230413  8.1282322
## 103   1.2759246  7.8230413  6.0001488
## 104   1.2759246  7.8230413  3.6230786
## 105   1.2759246  7.8230413  2.4165908
## 106   1.2759246  3.9885906 -1.2761191
## 107   1.2759246  3.9885906  7.1127273
## 108   1.2759246  3.9885906  1.0256657
## 109   1.2759246  3.9885906 15.8759113
## 110   1.2759246  3.9885906  6.6170191
## 111   1.2759246  3.9885906  5.9357163
## 112   1.2759246  3.9885906  7.1021421
## 113   1.2759246  3.9885906  8.1282322
## 114   1.2759246  3.9885906  6.0001488
## 115   1.2759246  3.9885906  3.6230786
## 116   1.2759246  3.9885906  2.4165908
## 117   1.2759246 -1.2761191  7.1127273
## 118   1.2759246 -1.2761191  1.0256657
## 119   1.2759246 -1.2761191 15.8759113
## 120   1.2759246 -1.2761191  6.6170191
## 121   1.2759246 -1.2761191  5.9357163
## 122   1.2759246 -1.2761191  7.1021421
## 123   1.2759246 -1.2761191  8.1282322
## 124   1.2759246 -1.2761191  6.0001488
## 125   1.2759246 -1.2761191  3.6230786
## 126   1.2759246 -1.2761191  2.4165908
## 127   1.2759246  7.1127273  1.0256657
## 128   1.2759246  7.1127273 15.8759113
## 129   1.2759246  7.1127273  6.6170191
## 130   1.2759246  7.1127273  5.9357163
## 131   1.2759246  7.1127273  7.1021421
## 132   1.2759246  7.1127273  8.1282322
## 133   1.2759246  7.1127273  6.0001488
## 134   1.2759246  7.1127273  3.6230786
## 135   1.2759246  7.1127273  2.4165908
## 136   1.2759246  1.0256657 15.8759113
## 137   1.2759246  1.0256657  6.6170191
## 138   1.2759246  1.0256657  5.9357163
## 139   1.2759246  1.0256657  7.1021421
## 140   1.2759246  1.0256657  8.1282322
## 141   1.2759246  1.0256657  6.0001488
## 142   1.2759246  1.0256657  3.6230786
## 143   1.2759246  1.0256657  2.4165908
## 144   1.2759246 15.8759113  6.6170191
## 145   1.2759246 15.8759113  5.9357163
## 146   1.2759246 15.8759113  7.1021421
## 147   1.2759246 15.8759113  8.1282322
## 148   1.2759246 15.8759113  6.0001488
## 149   1.2759246 15.8759113  3.6230786
## 150   1.2759246 15.8759113  2.4165908
## 151   1.2759246  6.6170191  5.9357163
## 152   1.2759246  6.6170191  7.1021421
## 153   1.2759246  6.6170191  8.1282322
## 154   1.2759246  6.6170191  6.0001488
## 155   1.2759246  6.6170191  3.6230786
## 156   1.2759246  6.6170191  2.4165908
## 157   1.2759246  5.9357163  7.1021421
## 158   1.2759246  5.9357163  8.1282322
## 159   1.2759246  5.9357163  6.0001488
## 160   1.2759246  5.9357163  3.6230786
## 161   1.2759246  5.9357163  2.4165908
## 162   1.2759246  7.1021421  8.1282322
## 163   1.2759246  7.1021421  6.0001488
## 164   1.2759246  7.1021421  3.6230786
## 165   1.2759246  7.1021421  2.4165908
## 166   1.2759246  8.1282322  6.0001488
## 167   1.2759246  8.1282322  3.6230786
## 168   1.2759246  8.1282322  2.4165908
## 169   1.2759246  6.0001488  3.6230786
## 170   1.2759246  6.0001488  2.4165908
## 171   1.2759246  3.6230786  2.4165908
## 172   4.9063860  0.9127462  4.7962633
## 173   4.9063860  0.9127462 -0.4960505
## 174   4.9063860  0.9127462  1.3199713
## 175   4.9063860  0.9127462  9.1003173
## 176   4.9063860  0.9127462  7.8230413
## 177   4.9063860  0.9127462  3.9885906
## 178   4.9063860  0.9127462 -1.2761191
## 179   4.9063860  0.9127462  7.1127273
## 180   4.9063860  0.9127462  1.0256657
## 181   4.9063860  0.9127462 15.8759113
## 182   4.9063860  0.9127462  6.6170191
## 183   4.9063860  0.9127462  5.9357163
## 184   4.9063860  0.9127462  7.1021421
## 185   4.9063860  0.9127462  8.1282322
## 186   4.9063860  0.9127462  6.0001488
## 187   4.9063860  0.9127462  3.6230786
## 188   4.9063860  0.9127462  2.4165908
## 189   4.9063860  4.7962633 -0.4960505
## 190   4.9063860  4.7962633  1.3199713
## 191   4.9063860  4.7962633  9.1003173
## 192   4.9063860  4.7962633  7.8230413
## 193   4.9063860  4.7962633  3.9885906
## 194   4.9063860  4.7962633 -1.2761191
## 195   4.9063860  4.7962633  7.1127273
## 196   4.9063860  4.7962633  1.0256657
## 197   4.9063860  4.7962633 15.8759113
## 198   4.9063860  4.7962633  6.6170191
## 199   4.9063860  4.7962633  5.9357163
## 200   4.9063860  4.7962633  7.1021421
## 201   4.9063860  4.7962633  8.1282322
## 202   4.9063860  4.7962633  6.0001488
## 203   4.9063860  4.7962633  3.6230786
## 204   4.9063860  4.7962633  2.4165908
## 205   4.9063860 -0.4960505  1.3199713
## 206   4.9063860 -0.4960505  9.1003173
## 207   4.9063860 -0.4960505  7.8230413
## 208   4.9063860 -0.4960505  3.9885906
## 209   4.9063860 -0.4960505 -1.2761191
## 210   4.9063860 -0.4960505  7.1127273
## 211   4.9063860 -0.4960505  1.0256657
## 212   4.9063860 -0.4960505 15.8759113
## 213   4.9063860 -0.4960505  6.6170191
## 214   4.9063860 -0.4960505  5.9357163
## 215   4.9063860 -0.4960505  7.1021421
## 216   4.9063860 -0.4960505  8.1282322
## 217   4.9063860 -0.4960505  6.0001488
## 218   4.9063860 -0.4960505  3.6230786
## 219   4.9063860 -0.4960505  2.4165908
## 220   4.9063860  1.3199713  9.1003173
## 221   4.9063860  1.3199713  7.8230413
## 222   4.9063860  1.3199713  3.9885906
## 223   4.9063860  1.3199713 -1.2761191
## 224   4.9063860  1.3199713  7.1127273
## 225   4.9063860  1.3199713  1.0256657
## 226   4.9063860  1.3199713 15.8759113
## 227   4.9063860  1.3199713  6.6170191
## 228   4.9063860  1.3199713  5.9357163
## 229   4.9063860  1.3199713  7.1021421
## 230   4.9063860  1.3199713  8.1282322
## 231   4.9063860  1.3199713  6.0001488
## 232   4.9063860  1.3199713  3.6230786
## 233   4.9063860  1.3199713  2.4165908
## 234   4.9063860  9.1003173  7.8230413
## 235   4.9063860  9.1003173  3.9885906
## 236   4.9063860  9.1003173 -1.2761191
## 237   4.9063860  9.1003173  7.1127273
## 238   4.9063860  9.1003173  1.0256657
## 239   4.9063860  9.1003173 15.8759113
## 240   4.9063860  9.1003173  6.6170191
## 241   4.9063860  9.1003173  5.9357163
## 242   4.9063860  9.1003173  7.1021421
## 243   4.9063860  9.1003173  8.1282322
## 244   4.9063860  9.1003173  6.0001488
## 245   4.9063860  9.1003173  3.6230786
## 246   4.9063860  9.1003173  2.4165908
## 247   4.9063860  7.8230413  3.9885906
## 248   4.9063860  7.8230413 -1.2761191
## 249   4.9063860  7.8230413  7.1127273
## 250   4.9063860  7.8230413  1.0256657
## 251   4.9063860  7.8230413 15.8759113
## 252   4.9063860  7.8230413  6.6170191
## 253   4.9063860  7.8230413  5.9357163
## 254   4.9063860  7.8230413  7.1021421
## 255   4.9063860  7.8230413  8.1282322
## 256   4.9063860  7.8230413  6.0001488
## 257   4.9063860  7.8230413  3.6230786
## 258   4.9063860  7.8230413  2.4165908
## 259   4.9063860  3.9885906 -1.2761191
## 260   4.9063860  3.9885906  7.1127273
## 261   4.9063860  3.9885906  1.0256657
## 262   4.9063860  3.9885906 15.8759113
## 263   4.9063860  3.9885906  6.6170191
## 264   4.9063860  3.9885906  5.9357163
## 265   4.9063860  3.9885906  7.1021421
## 266   4.9063860  3.9885906  8.1282322
## 267   4.9063860  3.9885906  6.0001488
## 268   4.9063860  3.9885906  3.6230786
## 269   4.9063860  3.9885906  2.4165908
## 270   4.9063860 -1.2761191  7.1127273
## 271   4.9063860 -1.2761191  1.0256657
## 272   4.9063860 -1.2761191 15.8759113
## 273   4.9063860 -1.2761191  6.6170191
## 274   4.9063860 -1.2761191  5.9357163
## 275   4.9063860 -1.2761191  7.1021421
## 276   4.9063860 -1.2761191  8.1282322
## 277   4.9063860 -1.2761191  6.0001488
## 278   4.9063860 -1.2761191  3.6230786
## 279   4.9063860 -1.2761191  2.4165908
## 280   4.9063860  7.1127273  1.0256657
## 281   4.9063860  7.1127273 15.8759113
## 282   4.9063860  7.1127273  6.6170191
## 283   4.9063860  7.1127273  5.9357163
## 284   4.9063860  7.1127273  7.1021421
## 285   4.9063860  7.1127273  8.1282322
## 286   4.9063860  7.1127273  6.0001488
## 287   4.9063860  7.1127273  3.6230786
## 288   4.9063860  7.1127273  2.4165908
## 289   4.9063860  1.0256657 15.8759113
## 290   4.9063860  1.0256657  6.6170191
## 291   4.9063860  1.0256657  5.9357163
## 292   4.9063860  1.0256657  7.1021421
## 293   4.9063860  1.0256657  8.1282322
## 294   4.9063860  1.0256657  6.0001488
## 295   4.9063860  1.0256657  3.6230786
## 296   4.9063860  1.0256657  2.4165908
## 297   4.9063860 15.8759113  6.6170191
## 298   4.9063860 15.8759113  5.9357163
## 299   4.9063860 15.8759113  7.1021421
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## 311   4.9063860  5.9357163  8.1282322
## 312   4.9063860  5.9357163  6.0001488
## 313   4.9063860  5.9357163  3.6230786
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## 319   4.9063860  8.1282322  6.0001488
## 320   4.9063860  8.1282322  3.6230786
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## 322   4.9063860  6.0001488  3.6230786
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## 666  -0.4960505  6.6170191  7.1021421
## 667  -0.4960505  6.6170191  8.1282322
## 668  -0.4960505  6.6170191  6.0001488
## 669  -0.4960505  6.6170191  3.6230786
## 670  -0.4960505  6.6170191  2.4165908
## 671  -0.4960505  5.9357163  7.1021421
## 672  -0.4960505  5.9357163  8.1282322
## 673  -0.4960505  5.9357163  6.0001488
## 674  -0.4960505  5.9357163  3.6230786
## 675  -0.4960505  5.9357163  2.4165908
## 676  -0.4960505  7.1021421  8.1282322
## 677  -0.4960505  7.1021421  6.0001488
## 678  -0.4960505  7.1021421  3.6230786
## 679  -0.4960505  7.1021421  2.4165908
## 680  -0.4960505  8.1282322  6.0001488
## 681  -0.4960505  8.1282322  3.6230786
## 682  -0.4960505  8.1282322  2.4165908
## 683  -0.4960505  6.0001488  3.6230786
## 684  -0.4960505  6.0001488  2.4165908
## 685  -0.4960505  3.6230786  2.4165908
## 686   1.3199713  9.1003173  7.8230413
## 687   1.3199713  9.1003173  3.9885906
## 688   1.3199713  9.1003173 -1.2761191
## 689   1.3199713  9.1003173  7.1127273
## 690   1.3199713  9.1003173  1.0256657
## 691   1.3199713  9.1003173 15.8759113
## 692   1.3199713  9.1003173  6.6170191
## 693   1.3199713  9.1003173  5.9357163
## 694   1.3199713  9.1003173  7.1021421
## 695   1.3199713  9.1003173  8.1282322
## 696   1.3199713  9.1003173  6.0001488
## 697   1.3199713  9.1003173  3.6230786
## 698   1.3199713  9.1003173  2.4165908
## 699   1.3199713  7.8230413  3.9885906
## 700   1.3199713  7.8230413 -1.2761191
## 701   1.3199713  7.8230413  7.1127273
## 702   1.3199713  7.8230413  1.0256657
## 703   1.3199713  7.8230413 15.8759113
## 704   1.3199713  7.8230413  6.6170191
## 705   1.3199713  7.8230413  5.9357163
## 706   1.3199713  7.8230413  7.1021421
## 707   1.3199713  7.8230413  8.1282322
## 708   1.3199713  7.8230413  6.0001488
## 709   1.3199713  7.8230413  3.6230786
## 710   1.3199713  7.8230413  2.4165908
## 711   1.3199713  3.9885906 -1.2761191
## 712   1.3199713  3.9885906  7.1127273
## 713   1.3199713  3.9885906  1.0256657
## 714   1.3199713  3.9885906 15.8759113
## 715   1.3199713  3.9885906  6.6170191
## 716   1.3199713  3.9885906  5.9357163
## 717   1.3199713  3.9885906  7.1021421
## 718   1.3199713  3.9885906  8.1282322
## 719   1.3199713  3.9885906  6.0001488
## 720   1.3199713  3.9885906  3.6230786
## 721   1.3199713  3.9885906  2.4165908
## 722   1.3199713 -1.2761191  7.1127273
## 723   1.3199713 -1.2761191  1.0256657
## 724   1.3199713 -1.2761191 15.8759113
## 725   1.3199713 -1.2761191  6.6170191
## 726   1.3199713 -1.2761191  5.9357163
## 727   1.3199713 -1.2761191  7.1021421
## 728   1.3199713 -1.2761191  8.1282322
## 729   1.3199713 -1.2761191  6.0001488
## 730   1.3199713 -1.2761191  3.6230786
## 731   1.3199713 -1.2761191  2.4165908
## 732   1.3199713  7.1127273  1.0256657
## 733   1.3199713  7.1127273 15.8759113
## 734   1.3199713  7.1127273  6.6170191
## 735   1.3199713  7.1127273  5.9357163
## 736   1.3199713  7.1127273  7.1021421
## 737   1.3199713  7.1127273  8.1282322
## 738   1.3199713  7.1127273  6.0001488
## 739   1.3199713  7.1127273  3.6230786
## 740   1.3199713  7.1127273  2.4165908
## 741   1.3199713  1.0256657 15.8759113
## 742   1.3199713  1.0256657  6.6170191
## 743   1.3199713  1.0256657  5.9357163
## 744   1.3199713  1.0256657  7.1021421
## 745   1.3199713  1.0256657  8.1282322
## 746   1.3199713  1.0256657  6.0001488
## 747   1.3199713  1.0256657  3.6230786
## 748   1.3199713  1.0256657  2.4165908
## 749   1.3199713 15.8759113  6.6170191
## 750   1.3199713 15.8759113  5.9357163
## 751   1.3199713 15.8759113  7.1021421
## 752   1.3199713 15.8759113  8.1282322
## 753   1.3199713 15.8759113  6.0001488
## 754   1.3199713 15.8759113  3.6230786
## 755   1.3199713 15.8759113  2.4165908
## 756   1.3199713  6.6170191  5.9357163
## 757   1.3199713  6.6170191  7.1021421
## 758   1.3199713  6.6170191  8.1282322
## 759   1.3199713  6.6170191  6.0001488
## 760   1.3199713  6.6170191  3.6230786
## 761   1.3199713  6.6170191  2.4165908
## 762   1.3199713  5.9357163  7.1021421
## 763   1.3199713  5.9357163  8.1282322
## 764   1.3199713  5.9357163  6.0001488
## 765   1.3199713  5.9357163  3.6230786
## 766   1.3199713  5.9357163  2.4165908
## 767   1.3199713  7.1021421  8.1282322
## 768   1.3199713  7.1021421  6.0001488
## 769   1.3199713  7.1021421  3.6230786
## 770   1.3199713  7.1021421  2.4165908
## 771   1.3199713  8.1282322  6.0001488
## 772   1.3199713  8.1282322  3.6230786
## 773   1.3199713  8.1282322  2.4165908
## 774   1.3199713  6.0001488  3.6230786
## 775   1.3199713  6.0001488  2.4165908
## 776   1.3199713  3.6230786  2.4165908
## 777   9.1003173  7.8230413  3.9885906
## 778   9.1003173  7.8230413 -1.2761191
## 779   9.1003173  7.8230413  7.1127273
## 780   9.1003173  7.8230413  1.0256657
## 781   9.1003173  7.8230413 15.8759113
## 782   9.1003173  7.8230413  6.6170191
## 783   9.1003173  7.8230413  5.9357163
## 784   9.1003173  7.8230413  7.1021421
## 785   9.1003173  7.8230413  8.1282322
## 786   9.1003173  7.8230413  6.0001488
## 787   9.1003173  7.8230413  3.6230786
## 788   9.1003173  7.8230413  2.4165908
## 789   9.1003173  3.9885906 -1.2761191
## 790   9.1003173  3.9885906  7.1127273
## 791   9.1003173  3.9885906  1.0256657
## 792   9.1003173  3.9885906 15.8759113
## 793   9.1003173  3.9885906  6.6170191
## 794   9.1003173  3.9885906  5.9357163
## 795   9.1003173  3.9885906  7.1021421
## 796   9.1003173  3.9885906  8.1282322
## 797   9.1003173  3.9885906  6.0001488
## 798   9.1003173  3.9885906  3.6230786
## 799   9.1003173  3.9885906  2.4165908
## 800   9.1003173 -1.2761191  7.1127273
## 801   9.1003173 -1.2761191  1.0256657
## 802   9.1003173 -1.2761191 15.8759113
## 803   9.1003173 -1.2761191  6.6170191
## 804   9.1003173 -1.2761191  5.9357163
## 805   9.1003173 -1.2761191  7.1021421
## 806   9.1003173 -1.2761191  8.1282322
## 807   9.1003173 -1.2761191  6.0001488
## 808   9.1003173 -1.2761191  3.6230786
## 809   9.1003173 -1.2761191  2.4165908
## 810   9.1003173  7.1127273  1.0256657
## 811   9.1003173  7.1127273 15.8759113
## 812   9.1003173  7.1127273  6.6170191
## 813   9.1003173  7.1127273  5.9357163
## 814   9.1003173  7.1127273  7.1021421
## 815   9.1003173  7.1127273  8.1282322
## 816   9.1003173  7.1127273  6.0001488
## 817   9.1003173  7.1127273  3.6230786
## 818   9.1003173  7.1127273  2.4165908
## 819   9.1003173  1.0256657 15.8759113
## 820   9.1003173  1.0256657  6.6170191
## 821   9.1003173  1.0256657  5.9357163
## 822   9.1003173  1.0256657  7.1021421
## 823   9.1003173  1.0256657  8.1282322
## 824   9.1003173  1.0256657  6.0001488
## 825   9.1003173  1.0256657  3.6230786
## 826   9.1003173  1.0256657  2.4165908
## 827   9.1003173 15.8759113  6.6170191
## 828   9.1003173 15.8759113  5.9357163
## 829   9.1003173 15.8759113  7.1021421
## 830   9.1003173 15.8759113  8.1282322
## 831   9.1003173 15.8759113  6.0001488
## 832   9.1003173 15.8759113  3.6230786
## 833   9.1003173 15.8759113  2.4165908
## 834   9.1003173  6.6170191  5.9357163
## 835   9.1003173  6.6170191  7.1021421
## 836   9.1003173  6.6170191  8.1282322
## 837   9.1003173  6.6170191  6.0001488
## 838   9.1003173  6.6170191  3.6230786
## 839   9.1003173  6.6170191  2.4165908
## 840   9.1003173  5.9357163  7.1021421
## 841   9.1003173  5.9357163  8.1282322
## 842   9.1003173  5.9357163  6.0001488
## 843   9.1003173  5.9357163  3.6230786
## 844   9.1003173  5.9357163  2.4165908
## 845   9.1003173  7.1021421  8.1282322
## 846   9.1003173  7.1021421  6.0001488
## 847   9.1003173  7.1021421  3.6230786
## 848   9.1003173  7.1021421  2.4165908
## 849   9.1003173  8.1282322  6.0001488
## 850   9.1003173  8.1282322  3.6230786
## 851   9.1003173  8.1282322  2.4165908
## 852   9.1003173  6.0001488  3.6230786
## 853   9.1003173  6.0001488  2.4165908
## 854   9.1003173  3.6230786  2.4165908
## 855   7.8230413  3.9885906 -1.2761191
## 856   7.8230413  3.9885906  7.1127273
## 857   7.8230413  3.9885906  1.0256657
## 858   7.8230413  3.9885906 15.8759113
## 859   7.8230413  3.9885906  6.6170191
## 860   7.8230413  3.9885906  5.9357163
## 861   7.8230413  3.9885906  7.1021421
## 862   7.8230413  3.9885906  8.1282322
## 863   7.8230413  3.9885906  6.0001488
## 864   7.8230413  3.9885906  3.6230786
## 865   7.8230413  3.9885906  2.4165908
## 866   7.8230413 -1.2761191  7.1127273
## 867   7.8230413 -1.2761191  1.0256657
## 868   7.8230413 -1.2761191 15.8759113
## 869   7.8230413 -1.2761191  6.6170191
## 870   7.8230413 -1.2761191  5.9357163
## 871   7.8230413 -1.2761191  7.1021421
## 872   7.8230413 -1.2761191  8.1282322
## 873   7.8230413 -1.2761191  6.0001488
## 874   7.8230413 -1.2761191  3.6230786
## 875   7.8230413 -1.2761191  2.4165908
## 876   7.8230413  7.1127273  1.0256657
## 877   7.8230413  7.1127273 15.8759113
## 878   7.8230413  7.1127273  6.6170191
## 879   7.8230413  7.1127273  5.9357163
## 880   7.8230413  7.1127273  7.1021421
## 881   7.8230413  7.1127273  8.1282322
## 882   7.8230413  7.1127273  6.0001488
## 883   7.8230413  7.1127273  3.6230786
## 884   7.8230413  7.1127273  2.4165908
## 885   7.8230413  1.0256657 15.8759113
## 886   7.8230413  1.0256657  6.6170191
## 887   7.8230413  1.0256657  5.9357163
## 888   7.8230413  1.0256657  7.1021421
## 889   7.8230413  1.0256657  8.1282322
## 890   7.8230413  1.0256657  6.0001488
## 891   7.8230413  1.0256657  3.6230786
## 892   7.8230413  1.0256657  2.4165908
## 893   7.8230413 15.8759113  6.6170191
## 894   7.8230413 15.8759113  5.9357163
## 895   7.8230413 15.8759113  7.1021421
## 896   7.8230413 15.8759113  8.1282322
## 897   7.8230413 15.8759113  6.0001488
## 898   7.8230413 15.8759113  3.6230786
## 899   7.8230413 15.8759113  2.4165908
## 900   7.8230413  6.6170191  5.9357163
## 901   7.8230413  6.6170191  7.1021421
## 902   7.8230413  6.6170191  8.1282322
## 903   7.8230413  6.6170191  6.0001488
## 904   7.8230413  6.6170191  3.6230786
## 905   7.8230413  6.6170191  2.4165908
## 906   7.8230413  5.9357163  7.1021421
## 907   7.8230413  5.9357163  8.1282322
## 908   7.8230413  5.9357163  6.0001488
## 909   7.8230413  5.9357163  3.6230786
## 910   7.8230413  5.9357163  2.4165908
## 911   7.8230413  7.1021421  8.1282322
## 912   7.8230413  7.1021421  6.0001488
## 913   7.8230413  7.1021421  3.6230786
## 914   7.8230413  7.1021421  2.4165908
## 915   7.8230413  8.1282322  6.0001488
## 916   7.8230413  8.1282322  3.6230786
## 917   7.8230413  8.1282322  2.4165908
## 918   7.8230413  6.0001488  3.6230786
## 919   7.8230413  6.0001488  2.4165908
## 920   7.8230413  3.6230786  2.4165908
## 921   3.9885906 -1.2761191  7.1127273
## 922   3.9885906 -1.2761191  1.0256657
## 923   3.9885906 -1.2761191 15.8759113
## 924   3.9885906 -1.2761191  6.6170191
## 925   3.9885906 -1.2761191  5.9357163
## 926   3.9885906 -1.2761191  7.1021421
## 927   3.9885906 -1.2761191  8.1282322
## 928   3.9885906 -1.2761191  6.0001488
## 929   3.9885906 -1.2761191  3.6230786
## 930   3.9885906 -1.2761191  2.4165908
## 931   3.9885906  7.1127273  1.0256657
## 932   3.9885906  7.1127273 15.8759113
## 933   3.9885906  7.1127273  6.6170191
## 934   3.9885906  7.1127273  5.9357163
## 935   3.9885906  7.1127273  7.1021421
## 936   3.9885906  7.1127273  8.1282322
## 937   3.9885906  7.1127273  6.0001488
## 938   3.9885906  7.1127273  3.6230786
## 939   3.9885906  7.1127273  2.4165908
## 940   3.9885906  1.0256657 15.8759113
## 941   3.9885906  1.0256657  6.6170191
## 942   3.9885906  1.0256657  5.9357163
## 943   3.9885906  1.0256657  7.1021421
## 944   3.9885906  1.0256657  8.1282322
## 945   3.9885906  1.0256657  6.0001488
## 946   3.9885906  1.0256657  3.6230786
## 947   3.9885906  1.0256657  2.4165908
## 948   3.9885906 15.8759113  6.6170191
## 949   3.9885906 15.8759113  5.9357163
## 950   3.9885906 15.8759113  7.1021421
## 951   3.9885906 15.8759113  8.1282322
## 952   3.9885906 15.8759113  6.0001488
## 953   3.9885906 15.8759113  3.6230786
## 954   3.9885906 15.8759113  2.4165908
## 955   3.9885906  6.6170191  5.9357163
## 956   3.9885906  6.6170191  7.1021421
## 957   3.9885906  6.6170191  8.1282322
## 958   3.9885906  6.6170191  6.0001488
## 959   3.9885906  6.6170191  3.6230786
## 960   3.9885906  6.6170191  2.4165908
## 961   3.9885906  5.9357163  7.1021421
## 962   3.9885906  5.9357163  8.1282322
## 963   3.9885906  5.9357163  6.0001488
## 964   3.9885906  5.9357163  3.6230786
## 965   3.9885906  5.9357163  2.4165908
## 966   3.9885906  7.1021421  8.1282322
## 967   3.9885906  7.1021421  6.0001488
## 968   3.9885906  7.1021421  3.6230786
## 969   3.9885906  7.1021421  2.4165908
## 970   3.9885906  8.1282322  6.0001488
## 971   3.9885906  8.1282322  3.6230786
## 972   3.9885906  8.1282322  2.4165908
## 973   3.9885906  6.0001488  3.6230786
## 974   3.9885906  6.0001488  2.4165908
## 975   3.9885906  3.6230786  2.4165908
## 976  -1.2761191  7.1127273  1.0256657
## 977  -1.2761191  7.1127273 15.8759113
## 978  -1.2761191  7.1127273  6.6170191
## 979  -1.2761191  7.1127273  5.9357163
## 980  -1.2761191  7.1127273  7.1021421
## 981  -1.2761191  7.1127273  8.1282322
## 982  -1.2761191  7.1127273  6.0001488
## 983  -1.2761191  7.1127273  3.6230786
## 984  -1.2761191  7.1127273  2.4165908
## 985  -1.2761191  1.0256657 15.8759113
## 986  -1.2761191  1.0256657  6.6170191
## 987  -1.2761191  1.0256657  5.9357163
## 988  -1.2761191  1.0256657  7.1021421
## 989  -1.2761191  1.0256657  8.1282322
## 990  -1.2761191  1.0256657  6.0001488
## 991  -1.2761191  1.0256657  3.6230786
## 992  -1.2761191  1.0256657  2.4165908
## 993  -1.2761191 15.8759113  6.6170191
## 994  -1.2761191 15.8759113  5.9357163
## 995  -1.2761191 15.8759113  7.1021421
## 996  -1.2761191 15.8759113  8.1282322
## 997  -1.2761191 15.8759113  6.0001488
## 998  -1.2761191 15.8759113  3.6230786
## 999  -1.2761191 15.8759113  2.4165908
## 1000 -1.2761191  6.6170191  5.9357163
## 1001 -1.2761191  6.6170191  7.1021421
## 1002 -1.2761191  6.6170191  8.1282322
## 1003 -1.2761191  6.6170191  6.0001488
## 1004 -1.2761191  6.6170191  3.6230786
## 1005 -1.2761191  6.6170191  2.4165908
## 1006 -1.2761191  5.9357163  7.1021421
## 1007 -1.2761191  5.9357163  8.1282322
## 1008 -1.2761191  5.9357163  6.0001488
## 1009 -1.2761191  5.9357163  3.6230786
## 1010 -1.2761191  5.9357163  2.4165908
## 1011 -1.2761191  7.1021421  8.1282322
## 1012 -1.2761191  7.1021421  6.0001488
## 1013 -1.2761191  7.1021421  3.6230786
## 1014 -1.2761191  7.1021421  2.4165908
## 1015 -1.2761191  8.1282322  6.0001488
## 1016 -1.2761191  8.1282322  3.6230786
## 1017 -1.2761191  8.1282322  2.4165908
## 1018 -1.2761191  6.0001488  3.6230786
## 1019 -1.2761191  6.0001488  2.4165908
## 1020 -1.2761191  3.6230786  2.4165908
## 1021  7.1127273  1.0256657 15.8759113
## 1022  7.1127273  1.0256657  6.6170191
## 1023  7.1127273  1.0256657  5.9357163
## 1024  7.1127273  1.0256657  7.1021421
## 1025  7.1127273  1.0256657  8.1282322
## 1026  7.1127273  1.0256657  6.0001488
## 1027  7.1127273  1.0256657  3.6230786
## 1028  7.1127273  1.0256657  2.4165908
## 1029  7.1127273 15.8759113  6.6170191
## 1030  7.1127273 15.8759113  5.9357163
## 1031  7.1127273 15.8759113  7.1021421
## 1032  7.1127273 15.8759113  8.1282322
## 1033  7.1127273 15.8759113  6.0001488
## 1034  7.1127273 15.8759113  3.6230786
## 1035  7.1127273 15.8759113  2.4165908
## 1036  7.1127273  6.6170191  5.9357163
## 1037  7.1127273  6.6170191  7.1021421
## 1038  7.1127273  6.6170191  8.1282322
## 1039  7.1127273  6.6170191  6.0001488
## 1040  7.1127273  6.6170191  3.6230786
## 1041  7.1127273  6.6170191  2.4165908
## 1042  7.1127273  5.9357163  7.1021421
## 1043  7.1127273  5.9357163  8.1282322
## 1044  7.1127273  5.9357163  6.0001488
## 1045  7.1127273  5.9357163  3.6230786
## 1046  7.1127273  5.9357163  2.4165908
## 1047  7.1127273  7.1021421  8.1282322
## 1048  7.1127273  7.1021421  6.0001488
## 1049  7.1127273  7.1021421  3.6230786
## 1050  7.1127273  7.1021421  2.4165908
## 1051  7.1127273  8.1282322  6.0001488
## 1052  7.1127273  8.1282322  3.6230786
## 1053  7.1127273  8.1282322  2.4165908
## 1054  7.1127273  6.0001488  3.6230786
## 1055  7.1127273  6.0001488  2.4165908
## 1056  7.1127273  3.6230786  2.4165908
## 1057  1.0256657 15.8759113  6.6170191
## 1058  1.0256657 15.8759113  5.9357163
## 1059  1.0256657 15.8759113  7.1021421
## 1060  1.0256657 15.8759113  8.1282322
## 1061  1.0256657 15.8759113  6.0001488
## 1062  1.0256657 15.8759113  3.6230786
## 1063  1.0256657 15.8759113  2.4165908
## 1064  1.0256657  6.6170191  5.9357163
## 1065  1.0256657  6.6170191  7.1021421
## 1066  1.0256657  6.6170191  8.1282322
## 1067  1.0256657  6.6170191  6.0001488
## 1068  1.0256657  6.6170191  3.6230786
## 1069  1.0256657  6.6170191  2.4165908
## 1070  1.0256657  5.9357163  7.1021421
## 1071  1.0256657  5.9357163  8.1282322
## 1072  1.0256657  5.9357163  6.0001488
## 1073  1.0256657  5.9357163  3.6230786
## 1074  1.0256657  5.9357163  2.4165908
## 1075  1.0256657  7.1021421  8.1282322
## 1076  1.0256657  7.1021421  6.0001488
## 1077  1.0256657  7.1021421  3.6230786
## 1078  1.0256657  7.1021421  2.4165908
## 1079  1.0256657  8.1282322  6.0001488
## 1080  1.0256657  8.1282322  3.6230786
## 1081  1.0256657  8.1282322  2.4165908
## 1082  1.0256657  6.0001488  3.6230786
## 1083  1.0256657  6.0001488  2.4165908
## 1084  1.0256657  3.6230786  2.4165908
## 1085 15.8759113  6.6170191  5.9357163
## 1086 15.8759113  6.6170191  7.1021421
## 1087 15.8759113  6.6170191  8.1282322
## 1088 15.8759113  6.6170191  6.0001488
## 1089 15.8759113  6.6170191  3.6230786
## 1090 15.8759113  6.6170191  2.4165908
## 1091 15.8759113  5.9357163  7.1021421
## 1092 15.8759113  5.9357163  8.1282322
## 1093 15.8759113  5.9357163  6.0001488
## 1094 15.8759113  5.9357163  3.6230786
## 1095 15.8759113  5.9357163  2.4165908
## 1096 15.8759113  7.1021421  8.1282322
## 1097 15.8759113  7.1021421  6.0001488
## 1098 15.8759113  7.1021421  3.6230786
## 1099 15.8759113  7.1021421  2.4165908
## 1100 15.8759113  8.1282322  6.0001488
## 1101 15.8759113  8.1282322  3.6230786
## 1102 15.8759113  8.1282322  2.4165908
## 1103 15.8759113  6.0001488  3.6230786
## 1104 15.8759113  6.0001488  2.4165908
## 1105 15.8759113  3.6230786  2.4165908
## 1106  6.6170191  5.9357163  7.1021421
## 1107  6.6170191  5.9357163  8.1282322
## 1108  6.6170191  5.9357163  6.0001488
## 1109  6.6170191  5.9357163  3.6230786
## 1110  6.6170191  5.9357163  2.4165908
## 1111  6.6170191  7.1021421  8.1282322
## 1112  6.6170191  7.1021421  6.0001488
## 1113  6.6170191  7.1021421  3.6230786
## 1114  6.6170191  7.1021421  2.4165908
## 1115  6.6170191  8.1282322  6.0001488
## 1116  6.6170191  8.1282322  3.6230786
## 1117  6.6170191  8.1282322  2.4165908
## 1118  6.6170191  6.0001488  3.6230786
## 1119  6.6170191  6.0001488  2.4165908
## 1120  6.6170191  3.6230786  2.4165908
## 1121  5.9357163  7.1021421  8.1282322
## 1122  5.9357163  7.1021421  6.0001488
## 1123  5.9357163  7.1021421  3.6230786
## 1124  5.9357163  7.1021421  2.4165908
## 1125  5.9357163  8.1282322  6.0001488
## 1126  5.9357163  8.1282322  3.6230786
## 1127  5.9357163  8.1282322  2.4165908
## 1128  5.9357163  6.0001488  3.6230786
## 1129  5.9357163  6.0001488  2.4165908
## 1130  5.9357163  3.6230786  2.4165908
## 1131  7.1021421  8.1282322  6.0001488
## 1132  7.1021421  8.1282322  3.6230786
## 1133  7.1021421  8.1282322  2.4165908
## 1134  7.1021421  6.0001488  3.6230786
## 1135  7.1021421  6.0001488  2.4165908
## 1136  7.1021421  3.6230786  2.4165908
## 1137  8.1282322  6.0001488  3.6230786
## 1138  8.1282322  6.0001488  2.4165908
## 1139  8.1282322  3.6230786  2.4165908
## 1140  6.0001488  3.6230786  2.4165908

mean_norm1   = matrix(apply(contoh_norm1, 1, mean))
mean_xbar1   = mean(mean_norm1)
var_xbar1    = var(mean_norm1)

n2           = 4
contoh_norm2 = urnsamples(populasi, size = 4, replace = F, ordered = F)
mean_norm2   = matrix(apply(contoh_norm2, 1, mean))
mean_xbar2   = mean(mean_norm2)
var_xbar2    = var(mean_norm2)

n3           = 15
contoh_norm3 = urnsamples(populasi, size = 15, replace = F, ordered = F)
mean_norm3   = matrix(apply(contoh_norm3, 1, mean))
mean_xbar3   = mean(mean_norm3)
var_xbar3    = var(mean_norm3)

hist(mean_norm1,main = paste("(n = 3)"),xlab = "xbar")
hist(mean_norm2,main = paste("(n = 4)"),xlab = "xbar")
hist(mean_norm3,main = paste("(n = 15)"),xlab = "xbar")

hasil        = data.frame("."=c("mean","varian"),"Populasi"=c(5,12),"n=3"=c(mean_xbar1,var_xbar1),"n=4"=c(mean_xbar2,var_xbar2),"n=15"=c(mean_xbar3,var_xbar3))
hasil

##        . Populasi      n.3      n.4      n.15
## 1   mean        5 4.809415 4.809415 4.8094152
## 2 varian       12 4.547044 3.207524 0.2672558

Kesimpulan:
- Berdasarkan output di atas, contoh acak yang diambil dari populasi dengan mean = $μ$ dan ragam = $σ^2$ , maka semakin besar ukuran contoh, mean dari $\bar{x}$ akan semakin mendekati $μ$ dan ragamnya semakin mendekati $\frac{σ^2}n$
- Dilihat dari histogramnya pun, semakin besar ukuran contoh, kurva mendekati sebaran normal.

Ketakbiasan Penduga Parameter

$\bar{x}$ adalah penduga tak bias bagi $\mu$ , jika E( $\bar{x}$ ) = $\mu$
$s^2$ adalah penduga tak bias bagi $\sigma^2$ , jika E( $s^2$ ) = $\sigma^2$
Maka dalam hal ini kita akan membuktikan apakah benar nilai harapan penduga parameter sama dengan nilai parameternya

Algoritme:

Tentukan sebaran yang akan digunakan
Ulangi sebanyak k kali

Bangkitkan n buah data dari sebaran yang sudah ditentukan
Hitung nilai $\bar{x}$ dan $s^2$

Hitung rata-rata dari $\bar{x}$ dan $s^2$ , kemudian bandingan dengan $\mu$ dan $\sigma^2$

Aplikasi di R

Penerapan ketakbiasan penduga (mean)

#POPULASI TERHINGGA

#1. Sebaran Normal 
library(probs)
set.seed(123)
n              = 10
populasi1      = rnorm(20)
mean_pop1      = mean(populasi1)
sampel_normal1 = urnsamples(populasi1, size = 10, replace = F, ordered = F)
mean_normal1   = matrix(apply(sampel_normal1, 1, mean))
median_normal1 = matrix(apply(sampel_normal1, 1, median))
harapan_mean_norm1     = mean(mean_normal1)
harapan_median_norm1   = mean(median_normal1)
 
#2. Sebaran Eksponensial
library(probs)
set.seed(123)
n              = 10
populasi2      = rexp(20)
mean_pop2      = mean(populasi2)
sampel_exp1    = urnsamples(populasi2, size = 10, replace = F, ordered = F)
mean_exp1      = matrix(apply(sampel_exp1, 1, mean))
median_exp1 = matrix(apply(sampel_exp1, 1, median))
harapan_mean_exp1     = mean(mean_exp1)
harapan_median_exp1   = mean(median_exp1)
 
#3. Uniform
library(probs)
set.seed(123)
n              = 10
populasi3      = runif(20)
mean_pop3      = mean(populasi3)
sampel_unif1   = urnsamples(populasi3, size = 10, replace = F, ordered = F)
mean_unif1     = matrix(apply(sampel_unif1, 1, mean))
median_unif1   = matrix(apply(sampel_unif1, 1, median))
harapan_mean_unif1     = mean(mean_unif1)
harapan_median_unif1   = mean(median_unif1)

hasil = data.frame("Hasil"=c("mean_populasi","harapan_mean_contoh","harapan_median_contoh"),"Sebaran Normal"=c(mean_pop1,harapan_mean_norm1,harapan_median_norm1),"Sebaran Eksponensial"=c(mean_pop2,harapan_mean_exp1,harapan_median_exp1),"Sebaran Seragam"=c(mean_pop3,harapan_mean_unif1,harapan_median_unif1))

hasil

##                   Hasil Sebaran.Normal Sebaran.Eksponensial Sebaran.Seragam
## 1         mean_populasi      0.1416238            0.8111726       0.5508084
## 2   harapan_mean_contoh      0.1416238            0.8111726       0.5508084
## 3 harapan_median_contoh      0.1174878            0.4931612       0.5504018

Kesimpulan :

Berdasarkan output di atas, dengan populasi terhingga maupun tak hingga serta tiga sebaran yang berbeda, nilai harapan median contoh tetap berbeda dengan $μ$ dan nilai harapan rataan contoh ( $\bar{x}$ ) mendekati sama (pada populasi tak hingga) bahkan sama persis dengan nilai parameter rataan populasi $μ$ (pada populasi terhingga) sehingga penduga tak bias bagi $μ$ adalah ( $\bar{x}$ )
Pada populasi terhingga, percontohan bersifat unik artinya tidak ada percontohan yang berulang sehingga dapat dipastikan kombinasi contoh hanya muncul satu kali sehingga nilai parameter dan nilai harapan penduga parameter yang tak bias sama persis.
Pada populasi tak hingga, percontohan yang terambil secara acak merupakan sebagian dari keseluruhan kemungkinan percontohan yang ada sehingga nilai parameter dan nilai harapan penduga parameter yang tak bias tidak sama persis, namun sangat mendekati.

Penerapan ketakbiasan penduga (Ragam)

# POPULASI TERHINGGA

#Sebaran Normal
set.seed(888)
n        = 10
populasi = rnorm(20) 
sigma2   = var(populasi)*(20-1)/20 #fungsi var pada R adalah varian contoh (penyebut n-1) sehingga perlu dikali (n-1)/n

library(probs)
sampel      = urnsamples(populasi, size = 10, replace = F, ordered = F)

## Pembagi (n-1)
s2.n1       = matrix(apply(sampel, 1, var))
E.s2.n1     = mean(s2.n1)

## Pembagi (n)
s2.n        = s2.n1*(10-1)/10
E.s2.n      = mean(s2.n)

#Sebaran Eksponensial
set.seed(888)
n           = 10
populasi2   = rexp(20) 
sigma2.exp  = var(populasi2)*(20-1)/20

library(probs)
sampel_exp     = urnsamples(populasi2, size = 10, replace = F, ordered = F)

## Pembagi (n-1)
s2.n1.exp   = matrix(apply(sampel_exp, 1, var))
E.s2.n1.exp = mean(s2.n1.exp)

## Pembagi (n)
s2.n.exp    = s2.n1.exp*(10-1)/10
E.s2.n.exp  = mean(s2.n.exp)


hasil = data.frame( "."  = c("ragam populasi","nilai harapan ragam contoh (n-1)","nilai harapan ragam contoh (n)"), 
                    "Sebaran Normal" = c(sigma2, E.s2.n1, E.s2.n),"Sebaran Eksponensial" = c(sigma2.exp, E.s2.n1.exp, E.s2.n.exp))
hasil

##                                  . Sebaran.Normal Sebaran.Eksponensial
## 1                   ragam populasi       1.298573             1.750903
## 2 nilai harapan ragam contoh (n-1)       1.366919             1.843056
## 3   nilai harapan ragam contoh (n)       1.230227             1.658750

Kesimpulan : - Berdasarkan output di atas, dengan skenario populasi terhingga dan dua sebaran yang berbeda, nilai harapan ragam contoh dengan penyebut $n−1$ harusnya lebih mendekati nilai parameter daripada nilai harapan ragam contoh dengan penyebut n. Hal ini menunjukkan bahwa penduga tak bias bagi ragam populasi $(σ^2)$ adalah $s^2$ dengan penyebut adalah $n−1$ , meskipun masih terdapat celah perbedaan (tidak 100% tak berbias).
- Pada populasi terhingga, percontohan bersifat unik artinya tidak ada percontohan yang berulang sehingga dapat dipastikan kombinasi contoh hanya muncul satu kali. Jika pada penduga mean nilai parameter dan statistik penduga tak bias sama persis, pada penduga ragam ini hal tersebut tidak berlaku karena perhitungan ragam populasi perlu dikali dengan faktor koreksi $\frac{(n−1)}n$ sedangkan perhitungan ragam contoh (dengan penyebut $n−1$ ) tidak perlu mengalikan dengan faktor koreksi.
- Pada populasi tak hingga, percontohan yang terambil secara acak merupakan sebagian dari keseluruhan kemungkinan percontohan yang ada. Namun, dalam hal ini nilai parameter ragam dan nilai harapan penduga tak biasnya tidak sama persis, hanya mendekati.

Selang Kepercayaan

Apa arti dari SK 95%?
- SK 95% bagi $\theta$ : Kita percaya 95% bahwa selang a sampai b memuat nilai parameter $\theta$ yang sebenarnya
- SK 95%: Jika kita melakukan 100 kali percontohan acak dan setiap percontohan acak dibuat selang kepercayaannya, maka dari 100 SK yang terbentuk, ada 95 SK yang mencakup parameter sedangkan sisanya sebanyak 5 SK tidak mencakup parameter.

Algoritme

Tentukan sebaran yang akan digunakan
Ulangi sebanyak k kali

Bangkitkan n buah data dari sebaran yang sudah ditentukan
Hitung nilai $\bar{x}$ dan $s^2$
Hitung $\sigma^2_{\bar{x}}$ dan buat selang kepercayaan $(1-\alpha)$ %

Hitung proporsi banyaknya selang kepercayaan yang memuat $μ$ , bandingkan dengan $(1-α)$

Aplikasi di R

n1     = 10
k      = 100 #ulangan
alpha  = 0.05
mu     = 50
std    = 10
set.seed(123)
sampel.norm1 = matrix(rnorm(n1*k,mu,std),k)
xbar.norm1   = apply(sampel.norm1,1,mean)
s.norm1      = apply(sampel.norm1,1,sd)
SE.norm1     = s.norm1/sqrt(n1)
z.norm1      = qnorm(1-alpha/2)
SK.norm1     = (xbar.norm1-z.norm1*SE.norm1 < mu & mu < xbar.norm1+z.norm1*SE.norm1)
x.norm1      = sum(SK.norm1)/k #proporsi banyaknya SK yang memuat mu

n2     = 30
k      = 100 #ulangan
alpha  = 0.05
mu     = 50
std    = 10
set.seed(123)
sampel.norm2 = matrix(rnorm(n2*k,mu,std),k)
xbar.norm2   = apply(sampel.norm2,1,mean)
s.norm2      = apply(sampel.norm2,1,sd)
SE.norm2     = s.norm2/sqrt(n2)
z.norm2      = qnorm(1-alpha/2)
SK.norm2     = (xbar.norm2-z.norm2*SE.norm2 < mu & mu < xbar.norm2+z.norm2*SE.norm2)
x.norm2      = sum(SK.norm2)/k #proporsi banyaknya SK yang memuat mu

n3     = 100
k      = 100 #ulangan
alpha  = 0.05
mu     = 50
std    = 10
set.seed(123)
sampel.norm3 = matrix(rnorm(n3*k,mu,std),k)
xbar.norm3   = apply(sampel.norm3,1,mean)
s.norm3      = apply(sampel.norm3,1,sd)
SE.norm3     = s.norm3/sqrt(n3)
z.norm3      = qnorm(1-alpha/2)
SK.norm3     = (xbar.norm3-z.norm3*SE.norm3 < mu & mu < xbar.norm3+z.norm3*SE.norm3)
x.norm3      = sum(SK.norm3)/k #proporsi banyaknya SK yang memuat mu

hasil = data.frame("n" =c(10,30,100),"Ketepatan SK Sebaran Normal"=c(x.norm1, x.norm2, x.norm3))
hasil

##     n Ketepatan.SK.Sebaran.Normal
## 1  10                        0.93
## 2  30                        0.93
## 3 100                        0.96

matplot(rbind (xbar.norm2-z.norm2*SE.norm2, xbar.norm2+z.norm2*SE.norm2), rbind(1:k,1:k), col=ifelse(SK.norm2,"blue","red"), type = "l", lty = 1,main='Selang Kepercayaan 95% (n=100)', xlab='SK', ylab='banyak ulangan')
abline(v=mu)

Gambar ini adalah hasil dari simulasi selang kepercayaan 95% untuk rata-rata populasi $(μ=50)$ dengan ukuran sampel $n=100$
Simulasi dilakukan sebanyak $k=100$ kali, sehingga ada 100 selang kepercayaan yang dihasilkan.
Tujuan gambar ini adalah untuk memvisualisasikan seberapa sering selang kepercayaan berhasil menangkap nilai rata-rata populasi $(μ)$
Garis vertikal di $x=50$ mengartikan nilai rata-rata populasi $(μ=50)$ . Selang kepercayaan yang berhasil menangkap $μ$ akan melintasi garis ini.
Garis Horizontal mengartikan bahwa setiap garis mewakili selang kepercayaan dari satu sampel. Jika garis tersebut melintasi garis vertikal di $x=50$ , artinya selang kepercayaan tersebut berhasil menangkap $\mu$
Semakin besar ukuran contoh (n), maka proporsi SK yang memuat nilai parameter semakin mendekati kebenaran (1-alpha)

# Interval Kepercayaan
library(car)

## Warning: package 'car' was built under R version 4.4.3

## Loading required package: carData

## Warning: package 'carData' was built under R version 4.4.3

data("Prestige")

# Menghitung rata-rata
m <- mean(Prestige$income)
m

## [1] 6797.902

# Menghitung standar error
p <- dim(Prestige)[1]
se <- sd(Prestige$income)/sqrt(p)
se

## [1] 420.4089

# Menghitung nilai kritis t
tval <- qt(0.975, df=p-1)

# Menghitung interval kepercayaan
cat(paste("KI: [", round(m-tval*se, 2),",",round(m+tval*se,2),"]"))

## KI: [ 5963.92 , 7631.88 ]

Artinya, dengan tingkat kepercayaan 95%, rata-rata pendapatan populasi berada dalam rentang 5963.92 hingga 7631.88.

?urnsamples

## starting httpd help server ... done

Estimasi Distribusi dan Parameter Model

Ratna Nur Mustika Sanusi

2025-03-06

Teorema Limit Pusat

Algoritma

Aplikasi di R

Hampiran Normal Terhadap Geometrik

Hampiran Normal Terhadap Eksponensial

Sebaran Percontohan Sebaran Normal

Ketakbiasan Penduga Parameter

Algoritme:

Aplikasi di R

Penerapan ketakbiasan penduga (mean)

Penerapan ketakbiasan penduga (Ragam)

Selang Kepercayaan

Algoritme

Aplikasi di R