Show that the unbiased estimator of the variance 2 from a sample of size n = 2 is one
Chapter 9, Problem 9.5-3(choose chapter or problem)
Show that the unbiased estimator of the variance 2 from a sample of size n = 2 is one half of the square of the difference of the two observations. Thus, show that, if a 2k design is replicated, say, with Xi1 and Xi2, i = 1, 2, ... , 2k, then the estimate of the common 2 is 1 2k+1 2 k i=1 (Xi1 Xi2) 2 = MS(E). Under the usual assumptions, this equation implies that each of 2k[A]2/MS(E), 2k[B]2/MS(E), 2k[AB]2/MS(E), and so on has an F(1, 2k) distribution under the null hypothesis. This approach, of course, would provide tests for the significance of the various effects, including interactions.
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