Pembahasan Soal Ujian Profesi Aktuaris
Institusi |
: |
Persatuan Aktuaris Indonesia (PAI) |
Mata Ujian |
: |
Metoda Statistika |
Periode Ujian |
: |
Mei 2018 |
Nomor Soal |
: |
26 |
SOAL
Sebuah regresi 2 variabel mengestimasi 100 titik
\({Y_i} = \alpha + \beta {X_i} + {\varepsilon _i}\)
ESS (error sum of squares) =10000
\(\sum\limits_{}^{} {X_i^2 = 5000} \)
Hitunglah standar error \(\widehat \beta {\rm{ }}\) dan \(\widehat \alpha ,\) yaitu \({s_{\widehat \beta }}\) dan \({s_{\widehat \alpha }}\)
- 143 dan 1.01
- 167 dan 1.21
- 182 dan 1.323
- 193 dan 1.433
- 21 dan 1.5
Diketahui |
n=100
ESS=10000
\(\sum\limits_{}^{} {X_i^2 = 5000} \) |
Rumus yang digunakan |
ESS
\({Y_i} = \alpha + \beta {X_i} + {\varepsilon _i}\)
Maka
\({s_{_{\widehat \alpha }}} = \sqrt {\frac{{\sum\limits_{}^{} {X_i^2} }}{{n\sum\limits_{}^{} {x_i^2} }}} \sigma \)
\({s_{_{\widehat \beta }}} = \frac{\sigma }{{\sqrt {\sum\limits_{}^{} {X_i^2} } }}\)
\({\widehat \sigma ^2} = \frac{{ESS}}{{n – 2}}\) |
Proses pengerjaan |
\({\widehat \sigma ^2} = \frac{{ESS}}{{n – 2}} = \frac{{10000}}{{98}}\)
\(\widehat \sigma = \sqrt {\frac{{10000}}{{98}}} \)
\({s_{_{\widehat \beta }}} = \frac{\sigma }{{\sqrt {\sum\limits_{}^{} {X_i^2} } }} = \frac{{\sqrt {\frac{{10000}}{{98}}} }}{{\sqrt {5000} }} = 0.1428 = 0.143\)
\({s_{_{\widehat \alpha }}} = \sqrt {\frac{{\sum\limits_{}^{} {X_i^2} }}{{n\sum\limits_{}^{} {x_i^2} }}} \sigma = \frac{{\sqrt {5000} }}{{\sqrt {(100)(5000)} }}\sqrt {\frac{{10000}}{{98}}} = 1.01\) |
Jawaban |
a. 0.143 dan 1.01 |