Pembahasan Soal Ujian Profesi Aktuaris
| Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian |
: |
Metoda Statistika |
| Periode Ujian |
: |
Mei 2017 |
| Nomor Soal |
: |
21 |
SOAL
Sebuah regresi 2 variabel digunakan untuk mencocokkan data berikut ini:
-
Hitunglah \(Cov\left( {\widehat \alpha ,\widehat \beta } \right)\)
- -1.77
- -1.85
- -1.93
- -2.01
- -2.09
| Diketahui |
|
| Rumus yang digunakan |
\(\sum {{x_i}{y_i}} = \sum {\left( {{X_i} – \overline X } \right)\left( {{Y_i} – \overline Y } \right)} \)
\(\sum {x_i^2 = } \sum {{{\left( {{X_i} – \overline X } \right)}^2}} \)
\(\sum {y_i^2 = } \sum {{{\left( {{Y_i} – \overline Y } \right)}^2}} \)
\(\sum {\widehat \varepsilon _i^2} = \sum {y_i^2} – \frac{{{{\left( {\sum {{x_i}{y_i}} } \right)}^2}}}{{\sum {x_i^2} }}\)
\(\widehat \sigma = \sqrt {\frac{{\sum {\widehat \varepsilon _i^2} }}{{n – 2}}} \)
\(Cov\left( {\widehat \alpha ,\widehat \beta } \right) = – \overline X \left( {\frac{{{\sigma ^2}}}{{\sum {x_i^2} }}} \right)\) |
| Proses pengerjaan |
| i |
\({X_i}\) |
\({Y_i}\) |
\({X_i} – \overline X \) |
\({Y_i} – \overline Y \) |
\({\left( {{X_i} – \overline X } \right)^2}\) |
\({\left( {{Y_i} – \overline Y } \right)^2}\) |
\(\left( {{X_i} – \overline X } \right)\left( {{Y_i} – \overline Y } \right)\) |
| 1 |
2 |
10 |
-4 |
0 |
16 |
0 |
0 |
| 2 |
5 |
6 |
-1 |
-4 |
1 |
16 |
4 |
| 3 |
8 |
11 |
2 |
1 |
4 |
1 |
2 |
| 4 |
9 |
13 |
3 |
3 |
9 |
9 |
9 |
| Rata-Rata |
6 |
10 |
|
|
|
|
|
| Jumlah |
|
|
0 |
0 |
30 |
26 |
15 |
\(\sum {\widehat \varepsilon _i^2} = \sum {y_i^2} – \frac{{{{\left( {\sum {{x_i}{y_i}} } \right)}^2}}}{{\sum {x_i^2} }}\)
\(= {\sum {\left( {{Y_i} – \overline Y } \right)} ^2} – \frac{{{{\left( {\sum {\left( {{X_i} – \overline X } \right)\left( {{Y_i} – \overline Y } \right)} } \right)}^2}}}{{\sum {{{\left( {{X_i} – \overline X } \right)}^2}} }}\)
\(= 26 – \frac{{{{15}^2}}}{{30}}\)
\(= 18.5\)
\(\widehat \sigma = \sqrt {\frac{{\sum {\widehat \varepsilon _i^2} }}{{n – 2}}} \)
\(= \sqrt {9.25} \)
\({\widehat \sigma ^2} = 9.25\)
\(Cov\left( {\widehat \alpha ,\widehat \beta } \right) = – \overline X \left( {\frac{{{\sigma ^2}}}{{\sum {x_i^2} }}} \right)\)
\(= – 6\left( {\frac{{9.25}}{{30}}} \right)\)
\(= – 1.85\) |
| Jawaban |
b. -1.85 |
