Probability Proportional to Size Sampling with ReplicationSuppose you are the Municipal Agriculturist in the Municipality of Quezon, Nueva Ecija, and your office is taskedby the Mayor to estimate the barangay level average number of carabaos and the total number of carabao in themunicipality as well. It was agreed that sample barangays drawn will be completely enumerated.Assume that Table 1 gives the barangay level data for the Municipality of Quezon based on the recent Census ofAgriculture (2012).Table 1. CA-2012 barangay level information (hypothetical data)Number of Number ofcarabaofarmLand areaNumber ofcarabaosBARANGAYNumber of Palay areahouseholds(ha.)(ha.)raisershouseholdsBertese (B)4938178896428845DonaLucia (DL)95200379239262Dulong Bayan (DB)1613208907386489IlogBaliwag (IB)4833187707368396Poblacion 12 (P)12698397135199Pulong Bahay (PB)5427120299205268SanAlejandro (SA)10449415735517573SanAndres1 (SA1)1128097265221268SanAndres2 (SA2)14481108264214289SanManuel (SMa)6525513917353380Santa Clara (SC)3327209433363414SantaRita (SR)7645282638429637Santo Cristo (SCR)362388173128205Santo TomasFeria (STF)1913123241309323SanMiguel (SMI)4728116452306345TRUE TOTAL8244932942770346015893TRUE MEANA. Using the information in Table 1, answer the following:1. How many barangays are there in the Municipality of Quezon? N =2. What is the proportion of number of farm households to number of households in the municipality?3. What is the proportion of total palay area to total land area in the municipality?4. What is the average number of carabaos reared by a carabao raiser in the municipality?B. Parameters to be estimated. Let Y = barangay level number of carabaos. Now, suppose you would liketo estimate the following parameters. Give the appropriate notation for these parameters that you aregoing to use (at the left of the equal sign) and the corresponding formula (at the right side of thedescription)).1.= barangay level mean number of carabaos2.= total number of carabaos in the municipality. C. Probability proportional to size sampling with replacement (PPSWR). Draw a sample of n = 5 barangaysusing PPSWR using one of the selection procedures.1. Select in your judgment, the most appropriate auxiliary variable to use. Let this be the variable X.Why did you choose this variable?2. Show the solution or document the selection of how the barangays included in your PPSWR samplewere obtained.3. XN=4. Summarize the information for your sample in Table 2.Table 2. PPSWR information for the sample barangays.SampleBarangayNumber of carabaosin the samplebarangayYiYour chosenauxiliary variablePiń EN XiZ₁=XXNXiD.Estimation. Using your sample data, and applying the PPSWR estimation procedure,1. Estimate the barangay level mean number of carabaos.2. Estimate the variance of your estimate in (1). State your assumption(s) if there are any.At the local government units (LGUs) municipality level, it is important to have information about totalproduction. Thus,3. Estimate the total number of carabaos in the municipality.4. Estimate the variance of your estimate in (3). State your assumption(s) if there are any.I

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