Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Anda mencocokkan model berikut dalam empat pengamatan:
\({Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {\varepsilon _i},\) \(i = 1,2,3,4\)Diberikan data sebagai berikut:
| \(i\) | \({X_{2i}}\) | \({X_{3i}}\) |
| 1 | -4 | -3 |
| 2 | -3 | 4 |
| 3 | 3 | -4 |
| 4 | 4 | 3 |
Estimasi least square dari \({\beta _3}\) dinyatakan sebagai \({\hat \beta _3} = \sum\nolimits_{i = 1}^4 {{w_i}{Y_i}} \)
Tentukan nilai dari \(\left( {{w_1};{w_2};{w_3};{w_4}} \right)\)
- \(\left( { – 0,08; – 0,06;0,06;0,08} \right)\)
- \(\left( { – 0,06;0,08; – 0,08;0,06} \right)\)
- \(\left( {0,06; – 0,08;0,08; – 0,06} \right)\)
- \(\left( { – 0,05;0,10; – 0,10;0,05} \right)\)
- \(\left( {0,05; – 0,10;0,10; – 0,05} \right)\)
| Diketahui | \({Y_i} = {\beta _1} + {\beta _2}{X_{2i}} + {\beta _3}{X_{3i}} + {\varepsilon _i},\) \(i = 1,2,3,4\) dengan data
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| Rumus yang digunakan | \({{\hat \beta }_3} = \sum\nolimits_{i = 1}^n {{w_i}{Y_i}} \) \(= \sum\nolimits_{i = 1}^n {\left[ {\frac{{\left( {{x_i} – \bar x} \right)}}{{\sum\nolimits_{i = 1}^n {{{\left( {{x_i} – \bar x} \right)}^2}} }}} \right]} {Y_i}\) | |||||||||||||||
| Proses pengerjaan | \({w_i} = \frac{{\left( {{x_{3i}} – {{\bar x}_3}} \right)}}{{\sum\nolimits_{i = 1}^n {{{\left( {{x_{3i}} – {{\bar x}_3}} \right)}^2}} }}\), sehingga \({\bar x_3} = \frac{{ – 3 + 4 – 4 + 3}}{4} = 0\) dan \(\sum\nolimits_{i = 1}^n {{{\left( {{x_{3i}} – {{\bar x}_3}} \right)}^2}} = {\left( { – 3} \right)^2} + {\left( 4 \right)^2} + {\left( { – 4} \right)^2} + {\left( 3 \right)^2} = 50\) diperoleh \({w_1} = \frac{{ – 3}}{{50}} = – 0,06\) \({w_2} = \frac{4}{{50}} = 0,08\) \({w_3} = \frac{{ – 4}}{{50}} = – 0,08\) \({w_4} = \frac{3}{{50}} = 0,06\) | |||||||||||||||
| Jawaban | b. \(\left( { – 0,06;0,08; – 0,08;0,06} \right)\) |