Two independent determinations of the resistivity of the same reel of copper wire are…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory

Measurement and Method, Ch Graphs, straight lines and least squares

Problem

Two independent determinations of the resistivity of the same reel of copper wire are reported: Method A (four-wire bridge): ρ1=(1.68±0.02)×10−8 Ω m\rho_1 = (1.68 \pm 0.02)\times10^{-8}\,\mathrm{\Omega\,m}; Method B (wire diameter and resistance): ρ2=(1.72±0.03)×10−8 Ω m\rho_2 = (1.72 \pm 0.03)\times10^{-8}\,\mathrm{\Omega\,m}. The two results are independent, having used different apparatus. ((a)) Show that the two results are consistent by computing the consistency ratio z=(ρ1−ρ2)/u(ρ1)2+u(ρ2)2z = (\rho_1-\rho_2)/\sqrt{u(\rho_1)^2+u(\rho_2)^2}. ((b)) Combine them into a weighted mean ρˉw\bar\rho_{\rm w} with its uncertainty u(ρˉw)=1/∑iwiu(\bar\rho_{\rm w}) = 1/\sqrt{\sum_i w_i}, where wi=1/u(ρi)2w_i = 1/u(\rho_i)^2.

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