Determining Optimum Strata Boundary Points on Stratified Random Sampling using Dynamic Programming (Case: Expenditure per Capita of East Java Province Year 2008)

Abstract

Optimum stratification is the method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation. In order to make the strata internally homogenous, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. This could be achieved effectively by having the distribution of the main study variable known and create strata by cutting the range of the distribution at suitable points. The problem of finding Optimum Strata Boundaries (OSB) is considered as the problem of determining Optimum Strata Widths (OSW). The problem is formulated as a Mathematical Programming Problem (MPP), which minimizes the variance of the estimated population parameter under Neyman allocation subject to the restriction that sum of the widths of all the strata is equal to the total range of the distribution. The distributions of the study variable are considered as continuous with standard normal density functions. The formulated MPPs, which turn out to be multistage decision problems, can then be solved using dynamic programming technique proposed by Bühler and Deutler (1975). After the counting process using C++ program received the width of each stratum. From these results the optimal boundary point can be determined for each stratum. For the two strata to get the optimal point on the boundary x1 = 0.002. For the formation of three strata obtained the optimal point on the boundary x1 = -0.546 and x2 = 0.552. For the formation of four strata obtained optimal boundary point is x1 = -0.869, x2 = 0.003 and x3 = 0.878. In forming five strata obtained optimal boundary point x1 = -1.096, x2 = -0.331, x3 = 0.339 and x4 = 1.107. The establishment of a total of six strata obtained the optimal point on the boundary x1 = -1.267, x2 = -0.569, x3 = 0.005, x4 = 0.579 and x5 = 1.281.

Penarikan contoh (sampling) dalam survei adalah suatu proses untuk memilih sebagian elemen dari suatu populasi dengan prosedur tertentu sehingga dapat digunakan untuk menduga parameter populasi. Untuk mencapai tujuan tersebut diperlukan metode penarikan contoh yang sesuai, salah satunya adalah penarikan contoh acak berlapis (stratified random sampling). Pada penarikan contoh acak berlapis, perlu diperhatikan peubah apa yang digunakan sebagai dasar pembentukan strata. Apabila peubah kualitatif digunakan untuk stratifikasi, pada umumnya pembentukan strata tidak terlalu mengalami masalah, misalnya berdasarkan tingkat pendidikan, pekerjaan dan jenis kelamin. Sebaliknya jika peubah yang digunakan untuk stratifikasi adalah peubah kuantitatif maka diperlukan teknik tertentu untuk menentukan batas antar strata. Masalah penentuan batas optimum strata ekuivalen dengan masalah menentukan lebar optimum strata yang diformulasikan sebagai masalah pemrograman matematika. Salah satunya adalah pendekatan pemrograman dinamik Bühler dan Deutler yaitu dengan meminimumkan ragam dari parameter populasi yang diduga berdasarkan alokasi Neyman dengan batasan bahwa jumlah lebar dari semua strata sama dengan total jarak dari sebaran peubah yang diteliti.