Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Mei 2017 |
| Nomor Soal |
: |
25 |
SOAL
Sebuah model regresi linear \({Y_i} = \alpha + \beta {X_i} + {\varepsilon _i}\) digunakan untuk mencocokkan data berikut ini:
Hitunglah estimasi heterocedasticity-consistent dari \(Var[\widehat \beta ]\)
- 0.031
- 0.042
- 0.053
- 0.064
- 0.075
| Diketahui |
model regresi linear \({Y_i} = \alpha + \beta {X_i} + {\varepsilon _i}\)
|
| 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 |
0 |
1 |
-4 |
-4 |
16 |
16 |
-0.17647 |
1.176471 |
1.384083 |
22.14533 |
| 2 |
3 |
2 |
-1 |
-3 |
1 |
3 |
3.705882 |
-1.70588 |
2.910035 |
2.910035 |
| 3 |
5 |
6 |
1 |
1 |
1 |
1 |
6.294118 |
-0.29412 |
0.086505 |
0.086505 |
| 4 |
8 |
11 |
4 |
6 |
16 |
24 |
10.17647 |
0.823529 |
0.678201 |
10.85121 |
| Total |
16 |
20 |
0 |
0 |
34 |
44 |
20 |
8.88E-16 |
5.058824 |
35.99308 |
| Mean |
4 |
5 |
|
|
|
|
|
|
|
|
\(\widehat \beta = \frac{{{S_{XY}}}}{{{S_{XX}}}} = \frac{{\sum\limits_{i = 1}^n {({X_i} – \overline X )({Y_i} – \overline Y )} }}{{\sum\limits_{i = 1}^n {{{({X_i} – \overline X )}^2}} }} = 1.294118\)
\({\rm{ }}\widehat \alpha = \overline Y – \widehat \beta \overline X = – 0.17647\)
\(Var[\widehat \beta ] = \frac{{{S_{XX}}\varepsilon _i^2}}{{{{({S_{XX}})}^2}}} = 0.031136\) |
| Jawaban |
a. 0.031 |
