Random Samples (2 problems)

Let be the mean of a random sample of size n₁ from a population with mean m₁ and variance s². Similarly let be the mean of a random sample of size n₂ from a population with mean m₂ and variance s². Finally let be the mean of a random sample of size n₃ from a population with mean m₃ and variance s² (so the populations have possibly different means but equal variances). We would like to estimate

q = (m₃-m₂) - (m₂-m₁)

a) Propose an estimator (a linear combination of , ) and find its variance as a multiple of s².

b) Suppose that we are planning to use your estimator from (a) in a study with n_T total observations to make inference on q through an a=0.05 two-sided t-test of H₀: q=0; the test will have n_T-3 degrees of freedom. Let n₁ = n_T/6, n₂ = 4n_T/6, n₃ = n_T/6. If we feel confident that s = 4, what total sample size is required for 80% power for the test if in fact q = 2? Show your work.