A semiconductor laboratory runs five statistical checks on an LED production batch.
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 semiconductor laboratory runs five statistical checks on an LED production batch.
((a)) A spectral emission model is fitted to N=20 wavelength measurements with p=4 free parameters, giving χ2=8.0. Using Q(χ2;ν)=e−χ2/2∑i=0ν/2−1(χ2/2)i/i!, find ν, the upper-tail p-value and the lower tail P(χν2≤8.0), and say what the lower tail suggests about the error bars.
((b)) Twenty-five independent defect-rate measurements (standard deviation σ=1 in the measurement's units) test H0:μ=0 against H1:μ=0.5 at size α=0.05 (z0.05=1.645). Give the Neyman-Pearson threshold and its power, using Φ(0.855)=0.8036.
((c)) A line-offset parameter, free to take either sign and zero under the null, is added to the yield model, giving −2lnΛ=3.84. Using Φ(1.96)=0.9750, find the p-value from Wilks' theorem.
((d)) No defective units are found in a small pilot batch. Find the 99.7 per cent Neyman upper limit on the mean number of expected defects.
((e)) In final testing, 48 of 50 LEDs from the batch pass specification. With a uniform prior on the pass rate 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)]).