Praktikum dan Latihan Minggu 4 - Estimasi Distribusi dan Parameter Model
Benigno Utama_2404220001
2026-03-09
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 μ dan ragam σ2, maka semakin besar ukuran contoh, sebaran dari x¯ akan semakin mendekati sebaran normal dengan mean μ dan ragam σ2n
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 4.5.2##
## Attaching package: 'probs'## The following objects are masked from 'package:base':
##
## intersect, setdiff, unionset.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")
menunjukkan bagaimana distribusi rata-rata sampel 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
## 300 4.9063860 15.8759113 8.1282322
## 301 4.9063860 15.8759113 6.0001488
## 302 4.9063860 15.8759113 3.6230786
## 303 4.9063860 15.8759113 2.4165908
## 304 4.9063860 6.6170191 5.9357163
## 305 4.9063860 6.6170191 7.1021421
## 306 4.9063860 6.6170191 8.1282322
## 307 4.9063860 6.6170191 6.0001488
## 308 4.9063860 6.6170191 3.6230786
## 309 4.9063860 6.6170191 2.4165908
## 310 4.9063860 5.9357163 7.1021421
## 311 4.9063860 5.9357163 8.1282322
## 312 4.9063860 5.9357163 6.0001488
## 313 4.9063860 5.9357163 3.6230786
## 314 4.9063860 5.9357163 2.4165908
## 315 4.9063860 7.1021421 8.1282322
## 316 4.9063860 7.1021421 6.0001488
## 317 4.9063860 7.1021421 3.6230786
## 318 4.9063860 7.1021421 2.4165908
## 319 4.9063860 8.1282322 6.0001488
## 320 4.9063860 8.1282322 3.6230786
## 321 4.9063860 8.1282322 2.4165908
## 322 4.9063860 6.0001488 3.6230786
## 323 4.9063860 6.0001488 2.4165908
## 324 4.9063860 3.6230786 2.4165908
## 325 0.9127462 4.7962633 -0.4960505
## 326 0.9127462 4.7962633 1.3199713
## 327 0.9127462 4.7962633 9.1003173
## 328 0.9127462 4.7962633 7.8230413
## 329 0.9127462 4.7962633 3.9885906
## 330 0.9127462 4.7962633 -1.2761191
## 331 0.9127462 4.7962633 7.1127273
## 332 0.9127462 4.7962633 1.0256657
## 333 0.9127462 4.7962633 15.8759113
## 334 0.9127462 4.7962633 6.6170191
## 335 0.9127462 4.7962633 5.9357163
## 336 0.9127462 4.7962633 7.1021421
## 337 0.9127462 4.7962633 8.1282322
## 338 0.9127462 4.7962633 6.0001488
## 339 0.9127462 4.7962633 3.6230786
## 340 0.9127462 4.7962633 2.4165908
## 341 0.9127462 -0.4960505 1.3199713
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## 989 -1.2761191 1.0256657 8.1282322
## 990 -1.2761191 1.0256657 6.0001488
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## 1015 -1.2761191 8.1282322 6.0001488
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## 1018 -1.2761191 6.0001488 3.6230786
## 1019 -1.2761191 6.0001488 2.4165908
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## 1030 7.1127273 15.8759113 5.9357163
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## 1032 7.1127273 15.8759113 8.1282322
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## 1041 7.1127273 6.6170191 2.4165908
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## 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
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## 1051 7.1127273 8.1282322 6.0001488
## 1052 7.1127273 8.1282322 3.6230786
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## 1054 7.1127273 6.0001488 3.6230786
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## 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
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## 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
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## 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
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## 1100 15.8759113 8.1282322 6.0001488
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## 1103 15.8759113 6.0001488 3.6230786
## 1104 15.8759113 6.0001488 2.4165908
## 1105 15.8759113 3.6230786 2.4165908
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## 1107 6.6170191 5.9357163 8.1282322
## 1108 6.6170191 5.9357163 6.0001488
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## 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 2.4165908mean_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 0.2672558Kesimpulan:
- Berdasarkan output di atas, contoh acak yang diambil dari populasi
dengan mean = μ dan ragam = σ2, maka semakin besar ukuran contoh, mean
dari x¯ akan semakin mendekati μ dan ragamnya semakin
mendekati σ2n
- Dilihat dari histogramnya pun, semakin besar ukuran contoh, kurva
mendekati sebaran normal.
Ketakbiasan Penduga Parameter
x¯ adalah penduga tak bias bagi μ, jika E(x¯) = μ
s2 adalah penduga tak bias bagi σ2, jika E(s2) = σ2
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 x¯ dan s2
- Hitung rata-rata dari x¯ dan s2, kemudian bandingan dengan μ dan σ2
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 0.5504018Kesimpulan :
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 (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 (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 1.658750Kesimpulan : - 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 (σ2) adalah s2 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 (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 θ: Kita percaya 95% bahwa selang a sampai b memuat nilai
parameter θ 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 x¯ dan s2
Hitung σ2x¯ dan buat selang kepercayaan (1−α)%
- 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 muhasil = 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 0.96matplot(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 μ
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 4.5.2## Loading required package: carData## Warning: package 'carData' was built under R version 4.5.2data("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.