You may submit on paper or electronically via Blackboard. Please make sure your name is on every page of the assignment, and it is clearly marked which question you are answering. Your response will be graded for correctness and clarity.
Concentrations of the pollutants aldrin and hexachlorobenzene (HCB) in nanograms per liter were measured in ten surface water samples, ten mid-depth water samples, and ten bottom samples from the Wolf River in Tennessee. The samples were taken downstream from an abandoned dump site previously used by the pesticide industry. The full data set can be found at http://www.biostat.umn.edu/~lynn/iid/wolf.river.dat. For this problem, we consider only HCB measurements taken at the bottom and the surface. The question of interest is whether the distribution of HCB concentration depends on the depth at which the measurement was taken. The data for this problem are given below.
Surface
Bottom
3.74
5.44
4.61
6.88
4.00
5.37
4.67
5.44
4.87
5.03
5.12
6.48
4.52
3.89
5.29
5.85
5.74
6.85
5.48
7.16
Assume the observations are independent normal random variables with unknown depth-specific means Qs and Qb and precisions Rs and Rb. Assume independent improper reference priors for the surface and bottom parameters:
𝑔.
This prior can be treated as the product of two normal-gamma priors with µs = µb = 0, ks = kb =0, as = ab = -1/2, and bs = bb = ∞. (These are not valid normal-gamma distributions, but you can use the usual Bayesian conjugate updating rule to find the posterior distribution.) Find the joint posterior distribution for the parameters (Qs, Rs, Qb, Rb). Find 90% posterior credible intervals for Qs, Qb, Rs, and Rb. Comment on your results.
Use direct Monte Carlo to sample 10,000 observations from the joint posterior distribution of (Θ$, Θ&,Ρ$, Ρ&). Use your Monte Carlo samples to estimate 90% posterior credible intervals for Qs, Qb, Rs, and Rb. Compare with the result of Problem 1.
Use your direct Monte Carlo sample to estimate the probability that the mean bottom concentration Qb is higher than the mean surface concentration Qs and to estimate the probability that the standard deviation Sb of the bottom concentrations is higher than the standard deviation Ss of the surface concentrations.
Comment on your analysis. What are your conclusions about the distributions of surface and bottom concentrations? Is the assumption of normality reasonable? Are the means different for surface and bottom? The standard deviations?
