Praktikum PSS week 4_Estimasi Distribusi dan Parameter Model
Rahajeng Septri Annisa
2026-03-12
| Teorema Limit Pusat |
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#Hampiran Normal Terhadap Geometrik
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 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
## 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
## 348 0.9127462 -0.4960505 15.8759113
## 349 0.9127462 -0.4960505 6.6170191
## 350 0.9127462 -0.4960505 5.9357163
## 351 0.9127462 -0.4960505 7.1021421
## 352 0.9127462 -0.4960505 8.1282322
## 353 0.9127462 -0.4960505 6.0001488
## 354 0.9127462 -0.4960505 3.6230786
## 355 0.9127462 -0.4960505 2.4165908
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## 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
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## 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
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## 858 7.8230413 3.9885906 15.8759113
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## 862 7.8230413 3.9885906 8.1282322
## 863 7.8230413 3.9885906 6.0001488
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## 890 7.8230413 1.0256657 6.0001488
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## 892 7.8230413 1.0256657 2.4165908
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## 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
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## 908 7.8230413 5.9357163 6.0001488
## 909 7.8230413 5.9357163 3.6230786
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## 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
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## 936 3.9885906 7.1127273 8.1282322
## 937 3.9885906 7.1127273 6.0001488
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## 939 3.9885906 7.1127273 2.4165908
## 940 3.9885906 1.0256657 15.8759113
## 941 3.9885906 1.0256657 6.6170191
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## 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
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## 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
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## 956 3.9885906 6.6170191 7.1021421
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## 958 3.9885906 6.6170191 6.0001488
## 959 3.9885906 6.6170191 3.6230786
## 960 3.9885906 6.6170191 2.4165908
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## 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
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## 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
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## 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
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## 1003 -1.2761191 6.6170191 6.0001488
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## 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
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## 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 2.4165908mean_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 0.2672558| Ketakbiasan Penduga Parameter |
|---|
| #Penerapan ketakbiasan penduga (mean) |
| ``` r #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 (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) |
| ``` r # 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 (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
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 0.96matplot(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 4.4.3## 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 ]#Artinya, dengan tingkat kepercayaan 95%, rata-rata pendapatan populasi berada dalam rentang 5963.92 hingga 7631.88.
?urnsamples## starting httpd help server ... done