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 |