Stratified Random Sampling – Statistik dan Parameter

Keterangan Notasi

\[ \sum_{h=1}^{L} N_h = N \]
\[ W_h = \frac{N_h}{N} \]
\[ \sum_{h=1}^{L} n_h = n \]
\[ f_h = \frac{n_h}{N_h} \]
  • yhi : Nilai karakteristik unit ke-i strata ke-h
  • N : Jumlah populasi
  • Nh : Jumlah populasi di strata ke-h
  • Wh : Penimbang strata ke-h (stratum weight)
  • n : Jumlah sampel
  • nh : Jumlah sampel di strata ke-h
  • fh : Fraksi sampling strata ke-h
  • L : Banyaknya subpopulasi atau strata

Populasi

Rata-rata

  • Rata-rata karakteristik populasi di strata ke-h
\[ \bar{Y} = \frac{1}{N_h} \sum_{i=1}^{N_h} Y_{hi} \]
  • Rata-rata karakteristik populasi
\[ \bar{Y} = \frac{1}{N} \sum_{h=1}^{L} \sum_{i=1}^{N_h} Y_{hi} \]

karena,

\[ \bar{Y_h} = \frac{\sum_{i=1}^{N_h}Y_{hi}}{N_h} \] \[ \sum_{i=1}^{N_h}Y_{hi} = N_h \bar{Y_{h}} \]

sehingga,

\[ \bar{Y} = \frac{1}{N} \sum_{h=1}^{L} N_h \bar{Y_{h}} \] \[ \bar{Y} = \sum_{h=1}^{L} W_h \bar{Y_{h}} \]

Varians

  • Varians karakteristik populasi di strata ke-h
\[ \sigma ^{2}_{h} = \frac{1}{N_h} \sum_{i=1}^{N_h} (Y_{hi} – \hat{Y_h})^2 \] \[ S ^{2}_{h} = \frac{1}{N_h – 1} \sum_{i=1}^{N_h} (Y_{hi} – \hat{Y_h})^2 \]
  • Varians karakteristik populasi
\[ \sigma ^{2} = \frac{1}{N} \sum_{i=1}^{N} (Y_{i} – \bar{Y})^2 \]

Sampel

Rata-rata

  • Estimasi rata-rata karakteristik di strata ke-h
\[ \bar{y}_h = \frac{1}{n_h} \sum_{i=1}^{n_h} y_{hi} \]
  • Estimasi rata-rata karakteristik populasi
\[ \bar{y}_{st} = \frac{1}{N} \sum_{h=1}^{L} N_h \bar{y}_h \] \[ \bar{y}_{st} = \sum_{h=1}^{L} W_h \bar{y}_h \]

Varians

  • Sampling varians dari estimasi rata-rata
\[ V(\bar{y}_{st}) = V\left ( \sum_{h=1}^{L} W_h \bar{y}_h \right ) \] \[ V(\bar{y}_{st}) = \sum_{h=1}^{L} W_h^{2} V(\bar{y}_h) \]
  • Jika masing-masing strata dilakukan penarikan sampel secara SRS WOR, maka:
\[ V(\bar{y}_{h}) = (1-f_h) \frac{S_h^2}{n_h} \] \[ V(\bar{y}_{st}) = \sum_{h=1}^{L} W_h^2 (1-f_h) \frac{S_h^2}{n_h} \]

Unbiased estimator dari Sh2 adalah sh2 sehingga unbiased estimator dari sampling varians adalah

\[ V(\bar{y}_{st}) = \sum_{h=1}^{L} W_h^2 (1-f_h) \frac{s_h^2}{n_h} \]

Total

  • Estimasi total karakteristik di strata ke-h
\[ \hat{Y}_h = \frac{N_h}{n_h} \sum_{i=1}^{n_h} y_{hi} = N_h \bar{y}_h \]
  • Varians estimasi total karakteristik di strata ke-h
\[ v(\hat{Y}_h) = N_h^2 v(\bar{y}_h) \]
  • Estimasi total karakteristik populasi
\[ \hat{Y}_{st} = N \bar{y}_{st} = \sum_{h=1}^{L} N_h \bar{y}_{h} \]
  • Varians estimasi total karakteristik
\[ v(\hat{Y}_{st}) = N^2 v(\bar{y}_{st}) \]

Materi Lengkap

Berikut adalah materi lainnya yang membahas mengenai Stratified Random Sampling.


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