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 |
