📋 Daftar Isi
Peluang Terpilih Sampel (Inclusion Probability)
Ketika mengambil satu unit sebagai sampel, peluang unit ke-i untuk terpilih sebagai sampel adalah sebagai berikut.
\[
p_i = \frac{1}{N}
\]
Misalkan kita mengambil sampel sebanyak n kali, maka peluang unit ke-i untuk terpilih dalam sampel (inclusion probability) adalah penjumlahan dari peluang terpilihnya unit tersebut pada pengambilan yang pertama, kedua, ketiga, dst. sampai dengan pengambilan ke-n.
SRS WR
\[
\pi_i = \frac{1}{N} + \frac{1}{N} + \cdots + \frac{1}{N}
\]
\[
\pi_i = \sum_{i=1}^{n} \frac{1}{N}
\]
\[
\pi_i = \frac{n}{N}
\]
SRS WOR
\[
\pi_i = \frac{1}{N} + \frac{N-1}{N} \cdot \frac{1}{N-1} + \cdots + \frac{N-n}{N} \frac{1}{N-n}
\]
\[
\pi_i = \sum_{i=1}^{n} \frac{1}{N}
\]
\[
\pi_i = \frac{n}{N}
\]
Estimasi
.tg-wrap {padding-botom: 20px;} .tg {text-align:center;vertical-align:top;border-collapse:collapse;border-color:#ccc;border-spacing:0;} .tg td{background-color:#fff;border-color:#ccc;border-style:solid;border-width:1px;color:#333; font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{background-color:#f0f0f0;border-color:#ccc;border-style:solid;border-width:1px;color:#333; font-family:Arial, sans-serif;font-size:14px;font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-wa1i{width: 35%;font-weight:bold;text-align:center;vertical-align:middle} .tg .tg-57iy{background-color:#f9f9f9;text-align:center;vertical-align:middle} .tg .tg-awal{width:30%;font-weight:bold;text-align:center;vertical-align:middle} .tg .tg-nrix{text-align:center;vertical-align:middle} .tg .tg-sh4c{text-align:center;vertical-align:top} .tg-sh4c .mjx-chtml {text-align: center !important;}| Nilai yang Diestimasi | SRS WR | SRS WOR |
|---|---|---|
| Rata-Rata | \[ \overline{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \] | |
| Varians Rata-Rata | \[ v(\overline{y}) = \frac{s^2}{n} \] | \[ v(\overline{y}) = \frac{N-n}{N} \cdot \frac{s^2}{n} \] |
| Total | \[ \widehat{Y} = \frac{N}{n} \sum_{i=1}^{n} y_i = N \overline{y} \] | |
| Varians Total | \[ v(\overline{Y}) = N^2 \cdot \frac{s^2}{n} \] | \[ v(\overline{Y}) = N^2 \cdot \frac{N-n}{N} \cdot \frac{s^2}{n} \] |
Pembuktian
Rata-Rata
Estimasi rata-rata y¯ adalah estimasi yang unbiased dari parameter Y¯
\[
E(\overline{y}) = E(\frac{1}{n} \sum_{i=1}^{n} y_i)
\]
\[
E(\overline{y}) = \frac{1}{n} \sum_{i=1}^{n} E(y_i)
\]
\[
E(\overline{y}) = \frac{1}{n} \sum_{i=1}^{n} (\sum_{i=1}^{n} p_i Y_i)
\]
\[
E(\overline{y}) = \frac{1}{n} \sum_{i=1}^{n} (\sum_{i=1}^{n} \frac{Y_i}{N})
\]
\[
E(\overline{y}) = \frac{1}{n} \sum_{i=1}^{n} \overline{Y}
\]
\[
E(\overline{y}) = \frac{1}{n} (n \cdot \overline{Y})
\]
\[
E(\overline{y}) = \overline{Y}
\]
Varians Rata-Rata
\[
V(\overline{y}) = E(\overline{y} – \overline{Y})^2
\]
\[
V(\overline{y}) = E[\frac{1}{n} \sum_{i=1}^{n} (y_i – \overline{Y})]^2
\]
\[
V(\overline{y}) = \frac{1}{n^2} E[\sum_{i=1}^{n} (y_i – \overline{Y})]^2
\]
\[
V(\overline{y}) = \frac{1}{n^2} E[\sum_{i=1}^{n} (y_i – \overline{Y})^2 + \sum_{i \neq j}^{n} (y_i – \overline{Y})(y_j – \overline{Y})]
\]
\[
V(\overline{y}) = \frac{1}{n^2} \sum_{i=1}^{n} E(y_i – \overline{Y})^2 + \frac{1}{n^2} \sum_{i \neq j}^{n} E(y_i – \overline{Y})(y_j – \overline{Y})
\]
\[
V(\overline{y}) = \frac{1}{n^2} \sum_{i=1}^{n} [\sum_{i=1}^{n} \frac{1}{N} (y_i – \overline{Y})^2] + \frac{1}{n^2} \sum_{i \neq j}^{n} [\frac{1}{N} \cdot \frac{1}{N-1} \sum_{i \neq j}^{N} (y_i – \overline{Y})(y_j – \overline{Y})]
\]
\[
V(\overline{y}) = \frac{1}{n^2} \sum_{i=1}^{n} \sigma^2 + \frac{1}{n^2} \cdot \frac{1}{N} \cdot \frac{1}{N-1} \cdot \sum_{i \neq j}^{n} [ [\sum_{i=1}^{N} (y_i – \overline{Y})]^2 – \sum_{i=1}^{N} (y_i – \overline{Y})^2]
\]
\[
V(\overline{y}) = \frac{1}{n^2} (n \cdot \sigma^2) + \frac{1}{n^2} \cdot \frac{1}{N} \cdot \frac{1}{N-1} \cdot \sum_{i \neq j}^{n} [-\sum_{i=1}^{N} (y_i – \overline{Y})^2]
\]
\[
V(\overline{y}) = \frac{\sigma^2}{n} – (\frac{1}{n^2} \cdot \frac{1}{N} \cdot \frac{1}{N-1} \cdot \sum_{i \neq j}^{n} \sigma^2)
\]
\[
V(\overline{y}) = \frac{\sigma^2}{n} – \frac{1}{n^2} \cdot \frac{1}{N} \cdot \frac{1}{N-1} \cdot n(n-1) \sigma^2
\]
\[
V(\overline{y}) = \frac{N-n}{N-1} \cdot \frac{\sigma^2}{n}
\]
\[
V(\overline{y}) = \frac{N-n}{N} \cdot \frac{S^2}{n} \rightarrow Rumus \: Varians \: SRS \: WOR
\]
\[
V(\overline{y}) = (1 – \frac{n}{N}) \cdot \frac{S^2}{n}
\]
\[
V(\overline{y}) = (1-f)\cdot \frac{S^2}{n}
\]
Keterangan Tambahan
\[
\frac{N-n}{N} \rightarrow Finite \: Population \: Corrections \: (FPC)
\]
\[
f = \frac{n}{N} \rightarrow Sampling \: Fraction
\]
Jika sampel diambil dengan SRS WR, maka yi dan yj akan saling statistically independent, sehingga:
\[
E(y_i – \overline{Y})(y_j – \overline{Y}) = 0
\]
\[
V(\overline{y}) = \frac{1}{n^2} \cdot n \cdot \sigma^2 = \frac{\sigma^2}{n} \rightarrow Rumus \: Varians \: SRS \: WR
\]
Total Karakteristik
\[
\widehat{Y} = N \cdot \overline{y}
\]
Penduga total tersebut adalah unbiased estimator untuk parameter Y, dapat dibuktikan:
\[
E(\widehat{Y}) = E(N \cdot \overline{y}) = N \cdot E(\overline{y}) = N \cdot \overline{Y} = Y
\]
Estimasi varians dari penduga total karakteristik.
\[
v(\widehat{Y}) = v(N \cdot \overline{y}) = N^2 \cdot \overline{y}
\]
SRS WR
\[
v(\widehat{Y}) = N^2 \cdot \frac{s^2}{n}
\]
SRS WOR
\[
v(\widehat{Y}) = N^2 \cdot \frac{N-n}{N} \cdot \frac{s^2}{n}
\]
Prosedur Pemilihan Sampel
Lottery Method
Dilakukan dengan cara pengacakan menggunakan lotre
Tabel Angka Random
Menggunakan tabel yang telah disediakan dengan metode berikut:
- Independent Choice Of Digits
- Pendekatan Sisa (Remainder Approach)
- Pendekatan hasil bagi (Quotient Approach)
Materi Lengkap
Berikut adalah materi lainnya yang membahas mengenai Simple Random Sampling.
