Estimasi Distribusi dan Parameter Model Menggunakan R
RASIKHAH HUKMA SHABIYYAH
2026-03-09
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, unionA. HAMPIRAN NORMAL 1. HAMPIRAN NORMAL TERHADAP GEOMETRI
par(mfrow=c(3,1))
library(probs)
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")
Hampiran normal terhadap geometri menunjukkan bagaimana distribusi
rata-rata sampel xbar dari distribusi normal 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")
Berdasarkan hasil grafik histogram diatas menunjukkan bahwa rata-rata
sampel dari data eksponensial akan mendekati distribusi normal ketika
ukuran sampel meningkat.
- 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")
Berdasarkan grafik histogram diatas menunjukkan bahwa semakin besar
ukuran sampel (n), maka sebaran rata-rata dan contoh acak yang berasal
dari sebaran geometrik, eksponensial, maupun uniform akan mendekati
sebaran normal.
B. CONTOH 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
## 342 0.9127462 -0.4960505 9.1003173
## 343 0.9127462 -0.4960505 7.8230413
## 344 0.9127462 -0.4960505 3.9885906
## 345 0.9127462 -0.4960505 -1.2761191
## 346 0.9127462 -0.4960505 7.1127273
## 347 0.9127462 -0.4960505 1.0256657
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## 1122 5.9357163 7.1021421 6.0001488
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## 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 2.4165908par(mfrow=c(3,1))
library(probs)
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 0.2672558Berdasarkan grafik histogram serta nilai mean dan varian diatas, diperoleh bahwa rata-rata sampel (xbar) merupakan estimator yang tidak bias terhadap mean populasi. Sedangkan apabila varians semakin kecil maka ukuran sampelnya akan meningkat. Hal ini menunjukkan bahwa semakin besar ukuran sampel, distribusi rata-rata sampel semakin mendekati distribusi normal sehingga estimasi parameter populasi akan lebih akurat dan stabil.
C. Ketakbiasan Penduga Parameter
- 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 0.5504018Berdasarkan hasil perbandingan mean populasi, harapan mean contoh (rata-rata sampel), dan harapan median contoh pada tiga jenis distribusi diperoleh bahwa nilai mean populasi dan harapan mean contoh pada ketiga distribusi memiliki hasil yang sama. Ini menunjukkan bahwa mean sampel merupakan estimator tak bias terhadap mean populasi. Sedangkan harapan median contoh memiliki hasil yang sedikit berbeda dari mean populasi sehingga median bukan estimator tak bias untuk mean populasi.
- 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 1.658750Berdasarkan hasil perbandingan ragam (varians) populasi dengan nilai harapan ragam contoh (sampel) menggunakan dua rumus berbeda, yaitu n-1 dan n pada dua distribusi: normal dan eksponensial diperoleh penduga ragam dengan pembagi n-1 menghasilkan nilai yang lebih mendekati ragam populasi, sehingga bersifat bias. Sedangkan penduga ragam dengan pembagi n cenderung menghasilkan nilai ragam yang lebih kecil dari ragam populasi, sehingga bersifat bias. Pola ini berlaku baik pada distribusi normal maupun eksponensial, sehingga penggunaan n-1 lebih tepat dalam estimasi ragam populasi dari sampel.
D. Selang Kepercayaan
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 0.96Berdasarkan hasil ketepatan selang pada sebaran normal kepercayaan menunjukkan bahwa semakin besar uuran sempel (n) maka ketepatan selang kepercayaannya cenderung lebih tinggi sehingga hasilnya kan lebih baik dan stabil.
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
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 4.4.3## Loading required package: carData## Warning: package 'carData' was built under R version 4.4.3data("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 ]Berdasarkan hasil interval kepercayaan pada data prestige dengan menggunakan tingkat kepercayaan sebesar 95% menunjukkan bahwa rata-rata pendapatan populasi berada dalam rentang 5963,92 sampai 7631,88