29 April, 2024

Pembahasan Ujian PAI: A50 – No. 27 – 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	:	27

SOAL

Diberikan model regresi dibawah ini

\({Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {\varepsilon _i}\)

diketahui

\(\sum\limits_{}^{} {X_{2i}^2 = 1200} \) \(\sum\limits_{}^{} {X_{3i}^2 = 2200} \) \(\sum\limits_{}^{} {X_{2i}^{}X_{3i}^{} = 2500} \) \({s^2} = 1000\)

Hitunglah \(\widehat {Cov}(\widehat {{\beta _2}},\widehat {{\beta _3}})\)

Diketahui	\(\sum\limits_{}^{} {X_{2i}^2 = 1200} \) \(\sum\limits_{}^{} {X_{3i}^2 = 2200} \) \(\sum\limits_{}^{} {X_{2i}^{}X_{3i}^{} = 2500} \) \({s^2} = 1000\)
Rumus yang digunakan	\(\widehat {Cov}(\widehat {{\beta _2}},\widehat {{\beta _3}}) = \frac{{ – {s^2}\frac{{\sum {{X_{2i}}{X_{3i}}} }}{{\sqrt {\sum {X_{2i}^2\sum {X_{3i}^2} } } }}}}{{(1 – {{(\frac{{\sum {{X_{2i}}{X_{3i}}} }}{{\sqrt {\sum {X_{2i}^2\sum {X_{3i}^2} } } }})}^2})(\sqrt {\sum {X_{2i}^2\sum {X_{3i}^2} } } )}}\)
Proses pengerjaan	\(\widehat {Cov}(\widehat {{\beta _2}},\widehat {{\beta _3}}) = \frac{{ – {s^2}\frac{{\sum {{X_{2i}}{X_{3i}}} }}{{\sqrt {\sum {X_{2i}^2\sum {X_{3i}^2} } } }}}}{{(1 – \frac{{\sum {{X_{2i}}{X_{3i}}} }}{{\sqrt {\sum {X_{2i}^2\sum {X_{3i}^2} } } }})(\sqrt {\sum {X_{2i}^2\sum {X_{3i}^2} } } )}}\) \(\widehat {Cov}(\widehat {{\beta _2}},\widehat {{\beta _3}}) = \frac{{ – 1000\frac{{2500}}{{\sqrt {(1200)(2200)} }}}}{{(1 – {{(\frac{{2500}}{{\sqrt {(1200)(2200)} }})}^2})\sqrt {(1200)(2200)} }} = 0.69252\)
Jawaban	b. 0.69252