Sampling theory and methods by S. Sampath

By S. Sampath

Sampling idea and strategies offers the theoretical points of "Sample Surveys" in a lucid shape for the advantage of either undergraduate and publish graduate scholars of information. It assumes little or no history in chance theory.The writer provides intimately a number of sampling schemes, together with uncomplicated random sampling, unequal chance sampling, and systematic, stratified, cluster, and multistage sampling. as well as sampling schemes, he additionally discusses numerous estimating equipment, comparable to ratio and regression estimators, and covers intimately using superpopulation models.Many solved theoretical difficulties are integrated into just about all the chapters. this option is helping readers gather invaluable talents to resolve theoretical difficulties all alone.

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A , where band care both smaller than N. For any r let us take n ~ c + 1 . N + r] mod N = r= i 1 This again contradicts our assumption that all elements of s( r, n ) are distinct. Hence the theorem. _ LY~. yj being theyN •= . I L[Yci - value of the jth unit in the circular systematic sampling corresponding to the random start i. 7 Systematic Sampling in Two Dimensions The linear systematic sampling can~ extended for two dimensions populations in a straightforward manner. Here it is assumed that the nmlcl population units are arranged in the form of ml rows , each containing nk units and it is planned Systematic Sampling Schemes 45 to select a systematic sample of mn units.

2 We have pointed out, a simple random sample of size n is obtained by drawing n random numbers one by one without replacing and considering the units with the corresponding labels. If the random numbers are drawn with replacement and the units corresponding to the drawn numbers is treated as sample, we obtain what is known as a "Simple Random Sampling With Replacement " sample (SRSWR). 8 Show that in simple random sampling with replacement (a) the sample mean y is unbiased for the population mean (b) V(y)=[N-l]s2 Nn Y Solution If Yi, i =I.

__ ~ r 2 . Y, -Y)2 =N2b2[r-(~+l) +( ~2 }2 +Na2 -2Na2 = N 2b 2[r-(k2 + 1)] 2 + N(k -1)<1 2 (usmg . 31) we get the average variance of the expansion estimator under linear systematic sampling as EM [V(fLSs J) =N2[ k:;l ]+N(k -1)2 Hence the solution. 2 Let s r denote the set of labels included in a circular systematic ins, for which r + jk S Nand s, be the complement set for which r + jk S N . +xto the n the unit with largest label and .!.. f --------------1 n 1(r) < n 2n(N -k) Solution Case (1) n 1(r) = n In this case the corrected sample meari can be Written as Systematic Sampling Schemes 5 t _=[1 Yc I I~ +- ~Yr~j-l)k +-Yr+(n-l)k n .

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