SAMPLE SURVEYS — DU B.Sc (hons) Statistics Previous Year Question Papers

(b) Prove that the systematic sampling will be more efficient than SRSWOR if the intra class correlation coefficient tr. N|,. (Notations have their usual rreaning). (8,7) ) ; 8. Write short rote : B.Sc. (Hons.) STATISTICS Semester III Dulation: J Hours (i) Stratified Random Sampling (ii) Probability and non-probability sarnpling (iii) Steps in conducting sample surveys (5x3) 2. Attempt six questions in all. (a) Define Simple random sampling. Derive the variance of the sample mean based on SRSWOR. Also. derive the unbiased estimator of this variance. (b) Define Difference estimator and derive from it the regression estimator. Also obtain the variance of regression estimator under first approximation. (b) From a simple random sample of size n drawn from N units by SRSWOR, a simple random sub-sample of ni units is duplicated and added to the original sample. Show that the mean based on (n + nl) units is an unbiased estimator of the population mean. Also, obtain its variance. How does it compare with the variance of the estimator based on n units only? (7,8) 7. (a) For a given cost function C = crnr+ c,n, derive Var(V*) under optimum allocation. Further, assume that a sampler has two strata with relative sizes P, and P, and equal population standard deviations S, and Sr. He prefers to use proportional allocation but does not wish to incur a substantial increase in variance as compared with optimum allocation. Show that (ignoring fpc): (a) lf (X, Y,) is the pair of the variables defined for var(v"t ) P,c1 + P2c2 (r, J., * e, J., )-every unit (i : 1,2,...,N) of the population and yn and in are the corresponding sample means of a Compute the sample sizes n,, n, in the 2 strata minimizing the total size of sample in each case under the following conditions (ignore the fpc): (i) The standard error of the estimated population mean is to be 0.1. (ii) The standard error of the difference between the two estimated stratum means is to be 0.1 . (iii) The standard error of the estimated mean of each stratum is to be 0.1. (6,9) (a) A population of N units contains Nrunits belonging to class A and N, (N-N,) units belonging to class B. Obtain an unbiased estimator of N, and unbiased estimator of its variance, under simple random sampling without rep lacement. simple random sample of size n selected without replacement, then prove that Hence prove that r( fn , y" ) = p(X,y). (Notations have their usual meaning). (b) Derive the condition under which a systematic sampling has the same precision as the corresponding stratified random sampling with one unit per stratum. (S,7) (a) Compare stratified random sampling under Neyman and proportional allocations with simple random sampling without replacement. (b) State the practical difficulties in adopting Neyman method of allocation of a sample to different strata'