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 |
