Question 5 The observed data y = {y₁j, i = 1,...,n, j = 1,...,m;} are the average results in an exam for school j within county i. The following hierarchical model is considered reasonable: Yij- Normal (₁,3), j = 1,...,mi μ₁~Normal (μc, o), i = 1,...,n. where μc, σs and σoc are unknown parameters which are each assigned a prior distribution. Suppose that we have generated a sample of size M from the joint posterior distribution pμc,σs,σc,μ1,...,pn | y). (a) Explain how to use the posterior sample to estimate the following: (i) the posterior mean for μc; (ii) a 95% credible interval for σs/oci (iii) the posterior probability that μ1 <μ2. (b) Explain how to generate a sample from the posterior predictive distribution of the result for a school not in our dataset, in each of the following two cases: (i) if the county containing the school is in our dataset; (ii) or if the county is not in our dataset.