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:
| \(Y\) |
3 |
9 |
14 |
| \(X\) |
1 |
4 |
10 |
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
| \(Y\) |
3 |
9 |
14 |
| \(X\) |
1 |
4 |
10 |
|
| 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 |
