Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Mei 2018 |
| Nomor Soal |
: |
25 |
SOAL
Sebuah model regresi linear \({Y_i} = 1 + \beta {X_i} + {\varepsilon _i}\) digunakan untuk mencocokkan data berikut ini:
Hitunglah estimasi heterocedasticity-consistent dari \(Var[\widehat \beta ]\)
- 0.0011
- 0.0015
- 0.0017
- 0.0019
- 0.0021
| Diketahui |
model regresi linear \({Y_i} = 1 + \beta {X_i} + {\varepsilon _i}\)
|
| Rumus yang digunakan |
\(\widehat \beta = \frac{{{S_{XY}}}}{{{S_{XX}}}} = \frac{{\sum\limits_{i = 1}^n {({X_i} – \overline X )({Y_i} – \overline Y )} }}{{\sum\limits_{i = 1}^n {{{({X_i} – \overline X )}^2}} }},\) dan \(\widehat \alpha = \overline Y – \widehat \beta \overline X \)
\({\varepsilon _i} = {Y_i} – {\widehat Y_i} = {Y_i} – (\widehat \alpha + \widehat \beta \overline X )\)
\(Var[\widehat \beta ] = \frac{{{S_{XX}}\varepsilon _i^2}}{{{{({S_{XX}})}^2}}}\) |
| Proses pengerjaan |
| I |
\({X_i}\) |
\({Y_i}\) |
\({X_i} – \overline X \) |
\({Y_i} – \overline Y \) |
\({S_{XX}}\) |
\({S_{XY}}\) |
\({\widehat Y_i}\) |
\({\varepsilon _i}\) |
\(\varepsilon _i^2\) |
\({S_{XX}}\varepsilon _i^2\) |
| 1 |
2 |
1 |
-2.66667 |
-2 |
7.111111 |
5.333333 |
1.285714 |
-0.28571 |
0.081633 |
0.580499 |
| 2 |
4 |
3 |
-0.66667 |
0 |
0.444444 |
0 |
2.571429 |
0.428571 |
0.183673 |
0.081633 |
| 3 |
8 |
5 |
3.333333 |
2 |
11.11111 |
6.666667 |
5.142857 |
-0.14286 |
0.020408 |
0.226757 |
| Total |
14 |
9 |
0 |
0 |
18.66667 |
12 |
9 |
-6.7E-16 |
0.285714 |
0.888889 |
| Mean |
4.666667 |
3 |
|
|
|
|
|
|
|
|
\(\widehat \beta = \frac{{{S_{XY}}}}{{{S_{XX}}}} = \frac{{\sum\limits_{i = 1}^n {({X_i} – \overline X )({Y_i} – \overline Y )} }}{{\sum\limits_{i = 1}^n {{{({X_i} – \overline X )}^2}} }} = 0.642857\)
\({\rm{ }}\widehat \alpha = \overline Y – \widehat \beta \overline X = 0\)
\(Var[\widehat \beta ] = \frac{{{S_{XX}}\varepsilon _i^2}}{{{{({S_{XX}})}^2}}} = 0.002551\) |
| Jawaban |
ANULIR |
