Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
November 2017 |
| Nomor Soal |
: |
23 |
SOAL
Sebuah regresi linear digunakan untuk mengestimasikan 10 titik \(\left( {{X_i},{Y_i}} \right)\). Estimasi \(\alpha \) adalah \(\hat \alpha \) dan estimasi \(\beta \) adalah \(\hat \beta \)
Diketahui pula:
- \(\sum {{{\left( {\hat \alpha + \hat \beta {X_i} – \bar Y} \right)}^2}} = 49\)
- Variansi sampel (sample variance) dari \(Y\) adalah 8
Hitunglah \(\sum {{{\left( {\hat \alpha + \hat \beta {X_i} – {Y_i}} \right)}^2}} \)
- 23
- 26
- 28
- 30
- 32
| Diketahui |
- \(\sum {{{\left( {\hat \alpha + \hat \beta {X_i} – \bar Y} \right)}^2}} = 49\)
- Variansi sampel (sample variance) dari \(Y\) adalah 8 dengan \(n = 10\)
|
| Rumus yang digunakan |
\(Var\left( {\hat Y} \right) = \frac{{\sum {{{\left( {{Y_i} – \bar Y} \right)}^2}} }}{{n – 1}}\)
\(= \frac{{\sum {{{\left( {{Y_i} – {{\hat Y}_i}} \right)}^2}} + \sum {{{\left( {{{\hat Y}_i} – \bar Y} \right)}^2}} }}{{n – 1}}\) |
| Proses pengerjaan |
\(Var\left( {\hat Y} \right) = \frac{{\sum {{{\left( {{Y_i} – {{\hat Y}_i}} \right)}^2}} + \sum {{{\left( {{{\hat Y}_i} – \bar Y} \right)}^2}} }}{{n – 1}}\)
\(= \frac{{\sum {{{\left( {\hat \alpha + \hat \beta {X_i} – {Y_i}} \right)}^2}} + \sum {{{\left( {\hat \alpha + \hat \beta {X_i} – \bar Y} \right)}^2}} }}{{n – 1}}\)
\(8 = \frac{{\sum {{{\left( {\hat \alpha + \hat \beta {X_i} – {Y_i}} \right)}^2}} + 49}}{9}\)
\(\sum {{{\left( {\hat \alpha + \hat \beta {X_i} – {Y_i}} \right)}^2}} = 72 – 49\)
\(= 23\) |
| Jawaban |
a. 23 |
