Solution
How Many Classrooms
Show the problem again
A district surveys students by choosing whole classrooms at random and measuring every student in each chosen room. Every classroom has exactly 16 students. Scores have a common variance $\sigma^2$, any two students in the same classroom have correlation $\rho = 0.2$, and students in different classrooms are independent. How many classrooms must be sampled so that the variance of the overall sample mean equals that of a simple random sample of 600 students drawn independently from the whole population?
Worked solution
A classroom mean of $m=16$ students has variance $\frac{\sigma^2}{m}\left(1 + (m-1)\rho\right)$, since the covariance terms add $m(m-1)\rho\sigma^2$ to the variance of the classroom total. The factor $1+(m-1)\rho = 1 + 15\cdot0.2 = 4$ is the design effect. With $k$ classrooms the sample mean has variance $\frac{4\sigma^2}{km}$. An SRS of 600 has variance $\sigma^2/600$. Equating, $km = 4\cdot 600 = 2400$ students, so $k = 2400/16 =$ **150** classrooms. The cluster design's effective sample size is only a quarter of its nominal size: 2400 students carry the information of 600.
Source: Design effect for one-stage cluster sampling, standard in Kish, Survey Sampling, and Cochran, Sampling Techniques. Statement written for AxiomIQ.