Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | Juni 2015 |
| Nomor Soal | : | 23 |
SOAL
Diketahui informasi sebagai berikut:
- \({y_i} = \beta {x_i} + {\varepsilon _i}\)
\(Var\left( {{\varepsilon _i}} \right) = {\left( {\frac{{{x_i}}}{2}} \right)^2}\)
| \(i\) | \({x_i}\) | \({y_i}\) |
| 1 | 1 | 7 |
| 2 | 2 | 5 |
| 3 | 3 | 2 |
| 4 | 4 | -3 |
Tentukan nilai estimasi weighted least square dari \(\beta \):
- 2,35
- 2,52
- 2,63
- 2,83
- 3,12
| Diketahui | - \({y_i} = \beta {x_i} + {\varepsilon _i}\)
\(Var\left( {{\varepsilon _i}} \right) = {\left( {\frac{{{x_i}}}{2}} \right)^2}\)
| \(i\) | \({x_i}\) | \({y_i}\) | | 1 | 1 | 7 | | 2 | 2 | 5 | | 3 | 3 | 2 | | 4 | 4 | -3 |
|
| Rumus yang digunakan | \({w_i} = \frac{1}{{{\sigma ^2}}} = \frac{1}{{Var\left( {{\varepsilon _i}} \right)}}\)
\(\hat \beta = \frac{{\sum\nolimits_i^n {{w_i}{x_i}{y_i}} }}{{\sum\nolimits_i^n {{w_i}x_i^2} }}\) |
| Proses pengerjaan | | \(i\) | \({x_i}\) | \({y_i}\) | \(Var\left( {{\varepsilon _i}} \right)\) | \({w_i}\) | \({w_i}{x_i}{y_i}\) | \(x_i^2\) | \({w_i}x_i^2\) | | 1 | 1.0000 | 7.0000 | 0.2500 | 4.0000 | 28.0000 | 1.0000 | 4.0000 | | 2 | 2.0000 | 5.0000 | 1.0000 | 1.0000 | 10.0000 | 4.0000 | 4.0000 | | 3 | 3.0000 | 2.0000 | 2.2500 | 0.4444 | 2.6667 | 9.0000 | 4.0000 | | 4 | 4.0000 | -3.0000 | 4.0000 | 0.2500 | -3.0000 | 16.0000 | 4.0000 | | Total | 10.0000 | 11.0000 | 7.5000 | 5.6944 | 37.6667 | 30.0000 | 16.0000 |
|
| \(\hat \beta = \frac{{\sum\nolimits_i^n {{w_i}{x_i}{y_i}} }}{{\sum\nolimits_i^n {{w_i}x_i^2} }}\)
\(\hat \beta = \frac{{37.6667}}{{16}}\)
\(\hat \beta = 2.35416667\) |
| Jawaban | a. 2,35 |
