Homework #1 (Due: Sept 26 2012) 1. Consider the followi

Homework #1 (Due: Sept 26 2012) 1. Consider the following sampling scheme: randomly permute the N units of a population. The sample a consists of the rst n units from the permuted list. Show that this leads to a SRS sampling scheme. 2. A SRS sample a1 of size n1 is selected from a population U of size N. Then another SRS sample a2 of size n2 is selected from a1. Show that a2 is a SRS of size n2 selected from U. 3. Select a SRS a1 of size n1 from a population U of size N. Then from U ?? a1 select a SRS a2 of size n2. The nal sample is a = a1 [ a2. Show that a is a SRS of size n selected from U. What is interesting about this result? 4. Consider the following sampling design for a population of size N = 3; U = f1; 2; 3g: Table 1: Sampling Design a p(a) a1 = (1; 2) 0.4 a2 = (1; 3) 0.3 a3 = (2; 3) 0.2 a4 = (1; 2; 3) 0.1 (a) Calculate all the rst and second order inclusion probabilities. (b) Find the value of E[n(a)] in two ways: (i) by a direct calculation using the denition; (ii) by use of the formula that expresses E[n(a)] as a function of the rst order inclusion probabilities. 5. A SRS sample a of n individuals is drawn from a frame that lists N individuals. The households corre- sponding to selected individuals are identied. Compute the inclusion probability of a household composed of M individuals where M 0: Think of a situation when such a sampling design could be useful.
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