1. Every sample of size 6 has equal chance (randomly choose 6 students out of 30). Answer following questions: (a) How many possible samples? We have a total of 30 students and want a sample of 6, therefore we must calculate how many combinations of 6 we can make out of 30, 30𝐶6 = 593,775 possible samples. (b) p(s) = ? The probability of selecting a specific sample is 1/593,775 = 0.000001684 1 𝑝(𝑠) = {153,775 0 𝑖𝑓 𝑠 ℎ𝑎𝑠 6 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (c) ns = ? The size of the sample is 𝑛𝑠 = 6 (d) Number of sample containing unit k? If we include unit k in the sample, we have 5 more spots to fill from the 29 students left. That is 29C5=118,755. (e) First order inclusion probabilities? The probability that unit k is included in the sample, 𝑃(𝑘 ∈ 𝑠), denoted as 𝜋𝑘 , is 𝜋𝑘 = 6 𝑓𝑜𝑟 𝑘 = 1,2, … 30 30 𝜋𝑘 = 1 𝑓𝑜𝑟 𝑘 = 1,2, … 30 5 (f) Second order inclusion probabilities? The probability that units k and l are included in the sample, 𝑃(𝑘, 𝑙 ∈ 𝑠), denoted as 𝜋𝑘𝑙 , is 𝜋𝑘𝑙 = 6(6 − 1) 𝑓𝑜𝑟 𝑘 ≠ 𝑙 = 1,2, … 30 30(30 − 1) 𝜋𝑘𝑙 = 30 𝑓𝑜𝑟 𝑘 ≠ 𝑙 = 1,2, … 30 870 𝜋𝑘𝑙 = 1 𝑓𝑜𝑟 𝑘 ≠ 𝑙 = 1,2, … 30 29 2. If the design 𝒑(. ) has a fixed sample size n, then 𝝅𝒊𝒌 ≥ 𝝅𝒊 + 𝝅𝒌 − 𝟏. We know that for simple size n from a population N, we ...
Purchase document to see full attachment