# Teorema limit pusat : Semakin banyak sampel acak dari suatu populasi, maka distribusi rata-rata sampel tersebut akan mendekati distribusi normal
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(1)
populasi = rgeom(20, 0.1) # membangkitkan data acak mengikuti distribusi geometrik, sebanyak 20 kali dengan peluang sukses 0.1
contoh_geo1 = urnsamples(populasi, size = 2, replace = F, ordered = F) # mencari kemungkinan kombinasi dengan cara mengambil 2 angka tanpa pengembalian dan tanpa memperhatikan urutan
mean_geo1 = matrix(apply(contoh_geo1, 1, mean)) # menghitung rata-rata kombinasi
contoh_geo2 = urnsamples(populasi, size = 5, replace = F, ordered = F)
mean_geo2 = matrix(apply(contoh_geo2, 1, mean))
contoh_geo3 = urnsamples(populasi, size = 10, replace = F, ordered = F)
mean_geo3 = matrix(apply(contoh_geo3, 1, mean))
par(mfrow=c(3,1))
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")
library(probs)
set.seed(22)
populasi = rexp(20)
contoh_exp1 = urnsamples(populasi, size = 2, replace = F, ordered = F)
mean_exp1 = matrix(apply(contoh_exp1, 1, mean))
contoh_exp2 = urnsamples(populasi, size = 5, replace = F, ordered = F)
mean_exp2 = matrix(apply(contoh_exp2, 1, mean))
contoh_exp3 = urnsamples(populasi, size = 10, replace = F, ordered = F)
mean_exp3 = matrix(apply(contoh_exp3, 1, mean))
par(mfrow=c(3,1))
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")
set.seed(3)
populasi = runif(20)
contoh_unif1 = urnsamples(populasi, size = 2, replace = F, ordered = F)
mean_unif1 = matrix(apply(contoh_unif1, 1, mean))
contoh_unif2 = urnsamples(populasi, size = 5, replace = F, ordered = F)
mean_unif2 = matrix(apply(contoh_unif2, 1, mean))
contoh_unif3 = urnsamples(populasi, size = 10, replace = F, ordered = F)
mean_unif3 = matrix(apply(contoh_unif3, 1, mean))
par(mfrow=c(3,1))
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 sampel, maka sebaran rata-rata dari sampel 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
library(probs)
set.seed(1299)
populasi = rnorm(20,5,sqrt(12))
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
## 356 0.9127462 1.3199713 9.1003173
## 357 0.9127462 1.3199713 7.8230413
## 358 0.9127462 1.3199713 3.9885906
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## 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
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## 780 9.1003173 7.8230413 1.0256657
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## 782 9.1003173 7.8230413 6.6170191
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## 786 9.1003173 7.8230413 6.0001488
## 787 9.1003173 7.8230413 3.6230786
## 788 9.1003173 7.8230413 2.4165908
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## 791 9.1003173 3.9885906 1.0256657
## 792 9.1003173 3.9885906 15.8759113
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## 794 9.1003173 3.9885906 5.9357163
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## 796 9.1003173 3.9885906 8.1282322
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## 804 9.1003173 -1.2761191 5.9357163
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## 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
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## 813 9.1003173 7.1127273 5.9357163
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## 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
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## 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
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## 830 9.1003173 15.8759113 8.1282322
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## 892 7.8230413 1.0256657 2.4165908
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## 918 7.8230413 6.0001488 3.6230786
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## 945 3.9885906 1.0256657 6.0001488
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## 947 3.9885906 1.0256657 2.4165908
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## 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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## 970 3.9885906 8.1282322 6.0001488
## 971 3.9885906 8.1282322 3.6230786
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## 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
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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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## 1026 7.1127273 1.0256657 6.0001488
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## 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
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## 1036 7.1127273 6.6170191 5.9357163
## 1037 7.1127273 6.6170191 7.1021421
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## 1044 7.1127273 5.9357163 6.0001488
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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
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## 1063 1.0256657 15.8759113 2.4165908
## 1064 1.0256657 6.6170191 5.9357163
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## 1067 1.0256657 6.6170191 6.0001488
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## 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
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## 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
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## 1087 15.8759113 6.6170191 8.1282322
## 1088 15.8759113 6.6170191 6.0001488
## 1089 15.8759113 6.6170191 3.6230786
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## 1092 15.8759113 5.9357163 8.1282322
## 1093 15.8759113 5.9357163 6.0001488
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## 1097 15.8759113 7.1021421 6.0001488
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## 1108 6.6170191 5.9357163 6.0001488
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## 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
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## 1122 5.9357163 7.1021421 6.0001488
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## 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)
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)
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)
par(mfrow=c(3,1))
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
# Berdasarkan output di atas, contoh acak yang diambil dari populasi dengan mean = μ dan ragam = σ^2, maka semakin besar ukuran contoh, mean dari x bar akan semakin mendekati μ dan ragamnya semakin mendekati σ^2/n. Dilihat dari histogramnya pun, semakin besar ukuran contoh, kurva mendekati sebaran normal.
# Ketakbiasan penduga parameter
# Penerapan ketakbiasan penduga (mean)
#1. Sebaran Normal
set.seed(4)
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
set.seed(5)
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
set.seed(6)
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 0.3762963 0.9007143 0.5644059
## 2 harapan_mean_contoh 0.3762963 0.9007143 0.5644059
## 3 harapan_median_contoh 0.2391035 0.7146673 0.5996101
# 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 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
# 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)
#Sebaran Normal
set.seed(7)
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
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(8)
populasi2 = rexp(20)
sigma2.exp = var(populasi2)*(20-1)/20
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 1.456025 0.5380908
## 2 nilai harapan ragam contoh (n-1) 1.532658 0.5664114
## 3 nilai harapan ragam contoh (n) 1.379392 0.5097702
# 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 (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
# Arti dari selang kepercayaan 95% adalah kita percaya 95% bahwa selang a sampai b memuat nilai parameter θ yang sebenarnya; dilakukan 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.
n1 = 10
k = 100 #ulangan
alpha = 0.05
mu = 50
std = 10
set.seed(9)
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(10)
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(11)
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 0.92
## 2 30 0.93
## 3 100 0.95
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)
# 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 4.4.3
data("Prestige")
m <- mean(Prestige$income)
m
## [1] 6797.902
p <- dim(Prestige)[1]
se <- sd(Prestige$income)/sqrt(p) # menghitung standar error
se
## [1] 420.4089
t_val <- qt(0.975, df=p-1) # menghitung nilai kritis t
cat(paste("KI: [", round(m-t_val*se, 2),",",round(m+t_val*se,2),"]")) # menghitung interval kepercayaan
## KI: [ 5963.92 , 7631.88 ]
# Artinya, dengan tingkat kepercayaan 95%, rata-rata pendapatan populasi berada dalam rentang 5963.92 hingga 7631.88.