Two independent laboratories measure the density of the same mineral sample by…

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

Mathematical Methods for Physics, Ch Estimation, fitting and hypothesis testing

Problem

Two independent laboratories measure the density of the same mineral sample by different methods. Laboratory A, using water displacement, reports

\rho_A = 2.702 \pm 0.004\ \mathrm{g\,cm^{-3}}

; laboratory B, using the lattice constant from X-ray diffraction, reports

\rho_B = 2.717 \pm 0.005\ \mathrm{g\,cm^{-3}}

, each a one-sigma uncertainty on an independent measurement. A two-sided test on the normal distribution uses the critical values

z = 1.96

at the

5

per cent level and

z = 2.58

at the

1

per cent level. ((a)) Compute the standardised difference

z

between the two reported densities. ((b)) State whether the difference is significant at the

5

per cent level, and separately at the

1

per cent level. ((c)) Name the type of error risked by declaring a disagreement when the densities are in fact equal, and the type risked by not declaring one when they truly differ, and say which applies to the decision reached at each level in part (b).

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