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 |
