Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2017 |
| Nomor Soal |
: |
20 |
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 |
9 |
| 2 |
2 |
3 |
| 3 |
3 |
4 |
| 4 |
4 |
-3 |
Tentukan nilai estimasi weighted least square dari \(\beta \), yaitu \(\hat \beta \) (dibulatkan 2 desimal).
- 2,62
- 2,69
- 2,77
- 2,85
- 2,93
| 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 |
9 |
| 2 |
2 |
3 |
| 3 |
3 |
4 |
| 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 |
9.0000 |
0.2500 |
4.0000 |
36.0000 |
1.0000 |
4.0000 |
| 2 |
2.0000 |
3.0000 |
1.0000 |
1.0000 |
6.0000 |
4.0000 |
4.0000 |
| 3 |
3.0000 |
4.0000 |
2.2500 |
0.4444 |
5.3333 |
9.0000 |
4.0000 |
| 4 |
4.0000 |
-3.0000 |
4.0000 |
0.2500 |
-3.0000 |
16.0000 |
4.0000 |
| Total |
10.0000 |
13.0000 |
7.5000 |
5.6944 |
44.3333 |
30.0000 |
16.0000 |
\(\hat \beta = \frac{{\sum\nolimits_i^n {{w_i}{x_i}{y_i}} }}{{\sum\nolimits_i^n {{w_i}x_i^2} }}\)
\(= \frac{{44,3333}}{{16}}\)
\(= 2,770833\) |
| Jawaban |
c. 2,77 |
