Question

A biochemist has been tasked with determining the amount of μ of a particular additive in a food product. In the laboratory there are two measuring instruments, A and B, which can be used for this. The measurement results from both instruments can be considered independent. In this connection, let X1,…, Xn denote n measurement results from instrument A and let Y1,…, Ym denote m measurement results from instrument B. We assume that the measurement results from instrument A and the measurement results from instrument B have different but known standard deviations σX = 1.3 and σY = 1.1, while the expectation value μ is common and unknown. The biochemist uses the following estimator for μ,

          μ = aY + bY = a/n Σ_i = 1, n, Xi + b/m Σi = 1, m, Yi,

where a and b are positive constants that the biochemist must choose depending on the sample sizes n and m.

Now assume that the biochemist for the same food sample does n = 10 measurements with instrument A and m = 14 measurements with instrument B. Find values ​​for a and b so that the estimator μ ^ becomes both expectation right and has as low variance as possible. What is the value of a?

Answer 1