A group measures a small dc voltage offset by taking independent readings

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 group measures a small dc voltage offset μ\mu by taking NN independent readings, each Gaussian with known standard deviation σ=4.0 mV\sigma = 4.0\ \mathrm{mV} and mean μ\mu. For this model the Cramer-Rao bound (Chapter 6) shows that the sample mean attains the minimum possible variance, var(μ^)=σ2/N\mathrm{var}(\hat\mu) = \sigma^2/N. ((a)) The group has time for N=35N=35 readings. Using this variance, find the smallest offset μ1\mu_1 for which the one-sided Neyman-Pearson test of H0:μ=0H_0:\mu=0 against H1:μ=μ1H_1:\mu=\mu_1, at significance α=0.05\alpha=0.05 (z0.05=1.645z_{0.05}=1.645), has power 1−β=0.901-\beta=0.90 (z0.10=1.282z_{0.10}=1.282). ((b)) Suppose the N=35N=35 readings are taken and yield a sample mean xˉ=1.9 mV\bar x = 1.9\ \mathrm{mV}. For Gaussian data of known σ\sigma, the likelihood-ratio statistic for H0:μ=0H_0:\mu=0 against the unrestricted best-fit alternative μ=xˉ\mu=\bar x is −2ln⁡Λ=Nxˉ2/σ2-2\ln\Lambda = N\bar x^2/\sigma^2. Compute −2ln⁡Λ-2\ln\Lambda and, using Wilks' theorem, the resulting p-value.

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