A search for a weak signal above a photomultiplier's dark-count background involves…
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory
Error, Chance and the Machine, Ch Tests, intervals and Bayesian inference
Problem
A search for a weak signal above a photomultiplier's dark-count background involves five calculations.
((a)) A background model is fitted to N=10 time bins with p=2 free parameters, giving χ2=9.0. Using Q(χ2;ν)=e−χ2/2∑i=0ν/2−1(χ2/2)i/i!, find ν and the p-value.
((b)) Sixteen independent rate measurements (standard deviation σ=1 in the fitted units) test H0:μ=0 against H1:μ=1 at size α=0.05 (z0.05=1.645). Give the Neyman-Pearson threshold and its power, using Φ(2.355)=0.9907.
((c)) A signal-rate parameter, which cannot be negative, is added to the background-only model, giving −2lnΛ=1.0. Using Φ(1.0)=0.8413, say whether Wilks' theorem applies as it stands, and find the p-value.
((d)) In a window where the expected dark-count background is 0.50 counts, one count is recorded. Find the 90 per cent Neyman upper limit on the mean signal count.
((e)) In a side test, the tube correctly registers 3 of 5 low-amplitude calibration pulses. With a uniform prior on the registration efficiency θ, give the posterior, its mean and mode, and an approximate central 68 per cent credible interval, treating the posterior as Gaussian with the same mean and variance (a Beta(a,b) density has variance ab/[(a+b)2(a+b+1)]).