Data Collectionhard
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A university estimates average GPA among 5,000 undergraduates using a two-stage sampling design: randomly select 10 departments, then 10 students per department (100 total). The within-department intraclass correlation for GPA is ρ = 0.08. The design effect is DEFF=1+(m1)ρ\text{DEFF} = 1 + (m-1)\rho where m is cluster size. Compared to a simple random sample of 100 students, how much larger a sample would two-stage sampling require to achieve the same precision?