Q

l. The quality of an assemny process is measured in defective items per day of production. Data was collected for a 40 consecutive days and are (read data left to right row I, then left to right row 2, etc): 21 9 1215 3 11 1011 1210 23 15 12 19 21 20 IS 9 20 24 19 9 21 13 26 23 1? 12 11 15 18 l? 10 22 9 2] 18 ll 12 14 The production manager is interested in assessing a possible changepoint in this process. Model the rst portion of the data as a PoisscnUL) model, the second portion of the data as a Poisson(cp) model, with the possible change occurring at day 1:. Assume a Bayesian model with independent priors: it ~ Uniform], 2, 3, .. ., 40} it ~ Gamma(2_0, 2) ip ~ Gamma(40, 2) Use a Gibbs sampler to generate samples from the posteriors of the three parameters with length of; say, 10000. (a) Determine an appropriate burnsin. (b) Discard the burn-in from the sample of 10000, then plot posterior densities (or histograms) for the three parameters. (c) Calculate summary statistics for the three parameters: mean. median, standard deviation, lower quartile, and upper quartile. (:1) Construct a 95% credible interval For the changepoint, k