Question: Let \[{{\bar{Y}}_{1}}\] be the mean of a random sample of size n ₁ from a population with mean  ₁ and variance  ² . Similarly let \[{{\bar{Y}}_{2}}\] be the mean of a random sample of size n ₂ from a population with mean  ₂ and variance  ² . Finally let \[{{\bar{Y}}_{3}}\] be the mean of a random sample of size n ₃ from a population with mean  ₃ and variance  ² (so the populations have possibly different means but equal variances). We would like to estimate

 _  _  _  _ 

1. Propose an estimator (a linear combination of \[{{\bar{Y}}_{1}}\] , \[{{\bar{Y}}_{2}}\] \[{{\bar{Y}}_{3}}\] ) and find its variance as a multiple of  ² .
2. Suppose that we are planning to use your estimator from (a) in a study with n _T total observations to make inference on  through an =0.05 two-sided t-test of H ₀ : =0; the test will have n _T -3 degrees of freedom. Let n ₁ = n _T /6, n ₂ = 4n _T /6, n ₃ = n _T /6. If we feel confident that  = 4, what total sample size is required for 80% power for the test if in fact  = 2? Show your work

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