Twenty independent water samples are each tested by a likelihood-ratio test for the…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods
Error, Chance and the Machine, Ch Tests, intervals and Bayesian inference
Problem
Twenty independent water samples are each tested by a likelihood-ratio test for the presence of a contaminant, comparing a null model with no contaminant against a model with one additional, unrestricted contamination-rate parameter. In one of the twenty samples the test statistic is −2lnΛ=4.7. A technician argues: "the χ12 distribution gives p≈0.03 for this sample, which is below 5 per cent, so this sample is significantly contaminated."
((a)) Compute the p-value of the single test from Wilks' theorem.
((b)) Compute the family-wise probability of at least one false alarm among all 20 tests, if each were run at a nominal 5 per cent significance level.
((c)) Compute the Bonferroni-corrected global p-value for the one significant-looking sample, and state whether the technician's claim is justified.