Question: Consider a variable with unknown mean \(\mu\) and variance \(\sigma^{2}\). We would like to sample to estimate \(\mu\), but have only a fixed amount of resources, $5,000, to do so.

Consider cluster sampling, where we sample \(M\) observations from each of \(K\) independent clusters. The reason this design is of interest for studying \(\mu\) is that while the first subject in each cluster is estimated to cost $\$ 50$ to sample, additional subjects from the same cluster cost only an additional $\$ 10$ per subject. Repeat subjects within a cluster are however positively correlated: \(\operatorname{Corr}\left(X_{i j}, X_{i k}\right)=\rho>0\) for \(j \neq k\).

We estimate \(\mu\) using the sample mean across subjects:

1. Show that
   \[\operatorname{Var}\left(\bar{X}_{C}\right)=\frac{\sigma^{2}}{K M}\{1+(M-1) \rho\}\]
2. Consider cluster sampling with \(M=6\). Under what conditions does \(\bar{X}_{\mathrm{C}}\) make more efficient use of the available $\$ 5,000$ than the sample mean of a simple random sample \(\left(\bar{X}_{\mathrm{SRS}}\right)\) for which each subject costs $\$ 50 ?
3. Assume that the within correlation is \(\rho=0.1\). Suggest an efficient survey design, that is, propose the optimal values of \(K\) and \(M\) given the budget restriction of $5,000. Include your working.

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