A count rate from a decaying source is measured in four equal time windows

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

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

Problem

A count rate from a decaying source is measured in four equal time windows, giving counts

C_j

at times

t_j

C=800

t=0\,\mathrm{s}

C=340

t=15\,\mathrm{s}

C=120

t=30\,\mathrm{s}

C=44

t=45\,\mathrm{s}

. Someone argues: “To get the half-life I fit

\ln C

against

t

. I used an ordinary, unweighted least-squares fit, because weighting the points seemed complicated and least squares just minimises the sum of squared residuals regardless. This gave a slope

b=-0.0650\,\mathrm{s}^{-1}\pm0.0019\,\mathrm{s}^{-1}

(one sigma), so the half-life is

\ln2/0.0650=10.7\pm0.3\,\mathrm{s}

.” ((a)) Identify the error in choosing an unweighted fit for these data, and state what the weight

w_j

of each point should be instead, with numerical values. ((b)) Redo the fit with those weights, and quote the slope and the half-life, each with its uncertainty. ((c)) Find the minimised

\chi^2

of the weighted fit and its number of degrees of freedom, and say whether Poisson errors account for the scatter of the four points.

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