Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | Agustus 2023 |
| Nomor Soal | : | 13 |
SOAL
Sebuah model \(Y_i = \beta_1 + \beta_2 X_{2i} + \beta_3 X_{3i} + \beta_4 X_{4i} + \varepsilon_i\)
i. \(N = 15\) dan \(ESS = 282{,}82\)
ii. \((X’X)^{-1} = \begin{bmatrix} 13{,}66 & -0{,}33 & 2{,}05 & -6{,}31 \\ -0{,}33 & -0{,}03 & 0{,}11 & 0{,}00 \\ 2{,}05 & 0{,}11 & 2{,}14 & -2{,}52 \\ -6{,}31 & 0{,}00 & -2{,}52 & 4{,}32 \end{bmatrix}\)
Hitunglah standard error dari \(\hat{\beta}_3 – \hat{\beta}_2\)
a. 6,4
b. 6,8
c. 7,1
d. 7,5
e. 7,8
| Diketahui | \(Y_i = \beta_1 + \beta_2 X_{2i} + \beta_3 X_{3i} + \beta_4 X_{4i} + \varepsilon_i\) i. \(N = 15\) dan \(ESS = 282{,}82\) ii. \((X’X)^{-1} = \begin{bmatrix} 13{,}66 & -0{,}33 & 2{,}05 & -6{,}31 \\ -0{,}33 & -0{,}03 & 0{,}11 & 0{,}00 \\ 2{,}05 & 0{,}11 & 2{,}14 & -2{,}52 \\ -6{,}31 & 0{,}00 & -2{,}52 & 4{,}32 \end{bmatrix}\) |
| Rumus yang digunakan | Formula,- \((X’X)^{-1} = \begin{bmatrix} \beta_{1,1} & \beta_{1,2} & \beta_{1,3} & \beta_{1,4} \\ \beta_{2,1} & \beta_{2,2} & \beta_{2,3} & \beta_{2,4} \\ \beta_{3,1} & \beta_{3,2} & \beta_{3,3} & \beta_{3,4} \\ \beta_{4,1} & \beta_{4,2} & \beta_{4,3} & \beta_{4,4} \end{bmatrix}\)
- \(Var(\hat{\beta}_3 – \hat{\beta}_2) = S^2 \left( \beta_{3,3} + \beta_{2,2} – 2\beta_{2,3} \right)\)
- \(S^2 = \dfrac{ESS}{n – k}\)
|
| Proses pengerjaan | - \(Var(\hat{\beta}_3 – \hat{\beta}_2) = \left( \dfrac{282{,}82}{15 – 4} \right) (2{,}14 + 0{,}03 – 2 \cdot 0{,}11)\)
\(Var(\hat{\beta}_3 – \hat{\beta}_2) = 50{,}136\)
\(\sigma_{\hat{\beta}_3 – \hat{\beta}_2} = 7{,}080687 \approx \mathbf{7{,}1}\) |
| Jawaban | c. 7,1 |
