28 April, 2024

Pembahasan Ujian PAI: A50 – No. 26 – Mei 2018

26 April 2019

by Agus Sofian Eka Hidayat, S.Pd., M.Ed., M.Sc.

with no comment

A50 - Metoda Statistika Aktuaria Edukasi Pembahasan Ujian Profesi Aktuaris (Persatuan Aktuaris Indonesia / PAI)

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 \({s_{\widehat \alpha }}\)

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