through school8.dat give weekly hours spent on homework for students sampled from eight different schools. Obtain posterior distributions for the true means for the eight different schools using a hierarchical normal model with the following prior parameters (cf. Section 8.4 in lecture notes): μ0 =7,γ02 =5,τ02 =10,η0 =2,σ02 =15,ν0 =2. (a) Run a Gibbs sampling algorithm to approximate the posterior distribution of {θ,σ2,μ,τ2}. Assess the convergence of the Markov chain, and find the effective sample size for {σ2,μ,τ2}. Run the chain long enough so that the effective sample sizes are all above 1,000. (b) Compute posterior means and 95% confidence regions for {σ2,μ,τ2}. Also, compare the posterior densities to the prior densities, and discuss what was learned from the data. (c) Plot the posterior density of R = τ2 , and compare it to a plot of the prior density of R. σ2+τ2 Describe the evidence for between-school variation.