Question: Suppose we are studying a trait that is distributed differently in two different groups in the population. We call these groups the "homogeneous" group (characterized by \(X_{\text {hom }}\) ) and the "heterogeneous" group (characterized by \(\left.X_{h e t}\right) . X_{h o m}\) and \(X_{h e t}\) have the following distributions:

The two distributions have the same mean \((\mu)\), but the variance of \(X_{h e t}\) is three times the variance of \(X_{\text {hom }}\). The parameter of interest is the overall population mean, \(\mu\). Assume that the probability of being in the homogenous group is 0.5 and the probability of being in the heterogeneous group is also 0.5.

1. Suppose we are able to conduct separate random samples from the homogenous group and the heterogeneous group of sizes \(n_{h o m}\) and \(n_{\text {het }}\). Consider this estimator for \(\mu\) :
   \[\hat{\mu}=0.5 \bar{X}_{h o m}+0.5 \bar{X}_{h e t}\]
   Find the expected value of this estimator, \(E[\hat{\mu}]\).
   Is \(\hat{\mu}\) an unbiased estimator of \(\mu\) ? Why or why not?
2. If \(n_{\text {het }} \neq n_{\text {hom }}\) is \(\hat{\mu}\) still unbiased? Why or why not?
3. Derive the variance of \(\hat{\mu}\).
4. You have enough money to sample 200 total people. Suppose you can choose between the two following sampling schemes:

Option A: \(n_{\text {hom }}=100, n_{\text {het }}=100\)

Option B: \(n_{\text {hom }}=70, n_{\text {het }}=130\)

Which sampling option would you choose? Justify your choice.

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