Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2016 |
| Nomor Soal | : | 24 |
SOAL
Diketahui informasi sebagai berikut
\({y_i} = \beta {x_i} + {\varepsilon _i}\)
- \(Var({\varepsilon _i}) = {\left( {\frac{{{x_i}}}{2}} \right)^2}\)
| \(i\) | \({x_i}\) | \({y_i}\) |
| 1 | 1 | 6 |
| 2 | 2 | 4 |
| 3 | 3 | 2 |
| 4 | 4 | -2 |
Tentukan nilai estimasi weighted least square dari \(\beta \)
- 1.35
- 1.88
- 1.96
- 2.04
- 2.35
| Diketahui | | \(i\) | \({x_i}\) | \({y_i}\) | | 1 | 1 | 6 | | 2 | 2 | 4 | | 3 | 3 | 2 | | 4 | 4 | -2 |
\({y_i} = \beta {x_i} + {\varepsilon _i}\)
\(Var({\varepsilon _i}) = {\left( {\frac{{{x_i}}}{2}} \right)^2}\) |
| Rumus yang digunakan | \({w_i} = \frac{1}{{{\sigma ^2}}} = \frac{1}{{Var({\varepsilon _i})}}\)
\(\widehat \beta = \frac{{\sum\limits_{i = 1}^n {{w_i}{x_i}{y_i}} }}{{\sum\limits_{i = 1}^n {{w_i}x_i^2} }}\) |
| Proses pengerjaan | | i | \({x_i}\) | \({y_i}\) | \(Var({\varepsilon _i})\) | \({w_i}\) | \({w_i}{x_i}{y_i}\) | \(x_i^2\) | \({w_i}x_i^2\) | | 1 | 1 | 6 | 0.25 | 4 | 24 | 1 | 4 | | 2 | 2 | 4 | 1 | 1 | 8 | 4 | 4 | | 3 | 3 | 2 | 2.25 | 0.444444 | 2.666667 | 9 | 4 | | 4 | 4 | -2 | 4 | 0.25 | -2 | 16 | 4 | | Total | 10 | 10 | 7.5 | 5.694444 | 32.66667 | 30 | 16 |
\(\widehat \beta = \frac{{\sum\limits_{i = 1}^n {{w_i}{x_i}{y_i}} }}{{\sum\limits_{i = 1}^n {{w_i}x_i^2} }} = \frac{{32.667}}{{16}} = 2.041\) |
| Jawaban | d. 2.04 |
