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.

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?