Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | November 2017 |
| Nomor Soal | : | 30 |
SOAL
Anda mengestimasikan model regresi linear \({Y_i} = \alpha + \beta {X_i} + {\varepsilon _i}\) berdasarkan data berikut ini:
Hitunglah estimasi heteroscedasticity-consistent dari \(Var\left[ {\hat \beta } \right]\) (dibulatkan 4 desimal).
- 0,0129
- 0,0139
- 0,0149
- 0,0159
- 0,0169
| Diketahui | model regresi linear \({Y_i} = \alpha + \beta {X_i} + {\varepsilon _i}\) dengan data |
| Rumus yang digunakan | \(\hat \beta = \frac{{{S_{XY}}}}{{{S_{XX}}}} = \frac{{\sum\nolimits_{i = 1}^n {\left( {{X_i} – \bar X} \right)\left( {{Y_i} – \bar Y} \right)} }}{{\sum\nolimits_{i = 1}^n {{{\left( {{X_i} – \bar X} \right)}^2}} }}\) dan \(\hat \alpha = \bar Y – \hat \beta \bar X\)
\({\varepsilon _i} = {Y_i} – {\hat Y_i} = {Y_i} – \left( {\hat \alpha + \hat \beta {X_i}} \right)\)
\(Var\left[ {\hat \beta } \right] = \frac{{{S_{XX}}\varepsilon _i^2}}{{{{\left( {{S_{XX}}} \right)}^2}}} = \frac{{\sum\nolimits_{i = 1}^n {{{\left( {{X_i} – \bar X} \right)}^2}\varepsilon _i^2} }}{{{{\left[ {\sum\nolimits_{i = 1}^n {{{\left( {{X_i} – \bar X} \right)}^2}} } \right]}^2}}}\) |
| Proses pengerjaan | | \(i\) | \({X_i}\) | \({Y_i}\) | \({X_i} – \bar X\) | \({Y_i} – \bar Y\) | \({S_{XX}}\) | \({S_{XY}}\) | \({\hat Y_i}\) | \({\varepsilon _i}\) | \(\varepsilon _i^2\) | \({S_{XX}}\varepsilon _i^2\) | | 1 | 1.00 | 3.00 | -4.00 | -5.67 | 16.00 | 22.67 | 4.00 | -1.00 | 1.00 | 16.00 | | 2 | 4.00 | 9.00 | -1.00 | 0.33 | 1.00 | -0.33 | 7.50 | 1.50 | 2.25 | 2.25 | | 3 | 10.00 | 14.00 | 5.00 | 5.33 | 25.00 | 26.67 | 14.50 | -0.50 | 0.25 | 6.25 | | Total | 15.00 | 26.00 | 0.00 | 0.00 | 42.00 | 49.00 | 26.00 | 0.00 | 3.50 | 24.50 | | Mean | 5.00 | 8.67 | | | | | | | | |
\(\hat \beta = \frac{{{S_{XY}}}}{{{S_{XX}}}} = \frac{{\sum\nolimits_{i = 1}^n {\left( {{X_i} – \bar X} \right)\left( {{Y_i} – \bar Y} \right)} }}{{\sum\nolimits_{i = 1}^n {{{\left( {{X_i} – \bar X} \right)}^2}} }}\)
\(= \frac{{49}}{{42}}\)
\(= 1,16667\)
\(\hat \alpha = \bar Y – \hat \beta \bar X\)
\(= 8,67 – 1,166667 \cdot 5\)
\(= 2,836665\)Diperoleh model estimasi regresi linear \({\hat Y_i} = \hat \alpha + \hat \beta {X_i} = 2,836665 + 1,166667{X_i}\)
\(Var\left[ {\hat \beta } \right] = \frac{{{S_{XX}}\varepsilon _i^2}}{{{{\left( {{S_{XX}}} \right)}^2}}} = \frac{{\sum\nolimits_{i = 1}^n {{{\left( {{X_i} – \bar X} \right)}^2}\varepsilon _i^2} }}{{{{\left[ {\sum\nolimits_{i = 1}^n {{{\left( {{X_i} – \bar X} \right)}^2}} } \right]}^2}}}\)
\(= \frac{{24,5}}{{{{\left( {42} \right)}^2}}}\)
\(= 0,013889\) |
| Jawaban | b. 0,0139 |
