A pendulum-timing test and a neutron-monitor count-rate measurement use the Wilks…

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

Error, Chance and the Machine, Ch Tests, intervals and Bayesian inference

Problem

A pendulum-timing test and a neutron-monitor count-rate measurement use the Wilks likelihood-ratio result and Bayesian conjugate updating. ((a)) (Wilks, an exact case.) Six independent period measurements xix_i (i=1,…,6i = 1,\ldots,6) of a pendulum are each Gaussian-distributed about the true period μ\mu with known standard deviation σ=0.020 s\sigma = 0.020\,\mathrm{s} (from the clock's calibration), and their mean is xˉ=2.017 s\bar x = 2.017\,\mathrm{s}. Derive the maximum-likelihood estimator μ^\hat\mu, and show that the likelihood-ratio statistic −2ln⁡Λ-2\ln\Lambda for H0:μ=μ0=2.000 sH_0: \mu = \mu_0 = 2.000\,\mathrm{s} against the unrestricted alternative equals exactly (xˉ−μ0σ/n)2\left(\dfrac{\bar x - \mu_0}{\sigma/\sqrt n}\right)^2 for any nn, not only as n→∞n\to\infty, and explain why this holds exactly rather than asymptotically for Gaussian data of known σ\sigma. Evaluate −2ln⁡Λ-2\ln\Lambda and its exact p-value from χ12\chi^2_1 for the numbers above. ((b)) (Conjugate posterior.) A neutron monitor counts cosmic-ray events at an unknown steady rate λ\lambda (events per hour). Starting from a Gamma(a0,b0)\mathrm{Gamma}(a_0, b_0) prior for λ\lambda with shape a0=2a_0 = 2 and rate b0=1 hourb_0 = 1\,\mathrm{hour}, derive from Bayes' theorem the posterior after observing nn counts in a total exposure TT hours, and show it is Gamma(a0+n, b0+T)\mathrm{Gamma}(a_0 + n,\, b_0 + T). The monitor records n=48n = 48 counts in T=10 hoursT = 10\,\mathrm{hours}. A Gamma(a,b)\mathrm{Gamma}(a, b) density with shape aa and rate bb has mean a/ba/b and variance a/b2a/b^2. Give the posterior mean, and, using the fact that a Gamma distribution with a large shape parameter is well approximated by a Gaussian of the same mean and variance, state an approximate central 68 per cent credible interval for λ\lambda.

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