Two surveys were independently conducted to estimate a

Question

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?