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Question: Two surveys were independently conducted to estimate a population mean, μ. Denote the estimates and their standard errors by X1 and X2 and σX1 and σX2 . Assume that X1 and X2 are unbiased. For some α and β, the two estimates can be combined to give a better estimator:
X = α X1¯ + βX2¯
a. Find the conditions on α and β that make the combined estimate unbiased.
b. What choice of α and β minimizes the variances, subject to the condition of unbiased-ness?
Suppose we have observations x1, x2, . . . , xn. Prove that the sum of deviations of the observations from their arithmetic mean is always equal to zero.
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