Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Metoda Statistika |
| Periode Ujian | : | Agustus 2023 |
| Nomor Soal | : | 11 |
SOAL
Anda diberikan sebuah data hasil studi kematian dengan data sensor kanan sebagai berikut:
| \(t_i\) | \(d_i\) | \(Y_i\) | \(\dfrac{d_i}{Y_i (Y_i – d_i)}\) | \(\widehat{S}(t_i)\) | \(\int_{t_i}^{\infty} \widehat{S}(t) \, dt\) |
| 1 | 15 | 100 | 0,0018 | 0,8500 | 14,424 |
| 8 | 20 | 65 | 0,0068 | 0,5885 | 8,472 |
| 17 | 13 | 40 | 0,0120 | 0,3972 | 3,178 |
| 25 | 31 | 31 | …….. | 0,0000 | 0,000 |
Tentukanlah selang kepercayaan 95% dari distribusi normal untuk rata-rata waktu bertahan hidup (mean survival time)
a. (13,5 : 17,4)
b. (13,7 : 17,2)
c. (13,9 : 17,0)
d. (14,1 : 16,8)
e. (14,3 : 16,6)
| Diketahui | tabel sensor:| \(t_i\) | \(d_i\) | \(Y_i\) | \(\dfrac{d_i}{Y_i (Y_i – d_i)}\) | \(\widehat{S}(t_i)\) | \(\int_{t_i}^{\infty} \widehat{S}(t) \, dt\) | | 1 | 15 | 100 | 0,0018 | 0,8500 | 14,424 | | 8 | 20 | 65 | 0,0068 | 0,5885 | 8,472 | | 17 | 13 | 40 | 0,0120 | 0,3972 | 3,178 | | 25 | 31 | 31 | …….. | 0,0000 | 0,000 |
|
| Rumus yang digunakan | Formula,- \(\hat{\mu}_t = \int_0^t \hat{S}(t)\, dt\)
- \(Var(\hat{\mu}_t) = \sum_{i=1}^{D} \left[ \int_{t_i}^{t} \hat{S}(t)\, dt \right]^2 \cdot \dfrac{d_i}{Y_i (Y_i – d_i)}\)
- Selang kepercayaan: \(\hat{\mu}_t \pm z_{\alpha} \cdot \sqrt{Var(\hat{\mu}_t)}\)
|
| Proses pengerjaan | - \(\hat{\mu}_t = \int_0^t \hat{S}(t)\, dt\)
\(\hat{\mu}_t = [1 \cdot (1)] + [0{,}85 \cdot (8-1)] + [0{,}5885 \cdot (17-8)] + [0{,}3972 \cdot (25-17)]\)
\(\hat{\mu}_t = 15{,}4241\)
- \(Var(\hat{\mu}_t) = (14{,}424)^2 \cdot 0{,}0018 + (8{,}472)^2 \cdot 0{,}0068 + (3{,}178)^2 \cdot 0{,}012\)
\(Var(\hat{\mu}_t) = 0{,}983758\)
- Selang kepercayaan \(15{,}4241 \pm 1{,}96 \cdot \sqrt{0{,}983758} = (13{,}48 \; ; \; 17{,}37)\)
|
| Jawaban | a. (13,5 : 17,4) |
