Rules: You must unite your name and the East .5 digits of your student ID on the submitted work. Suppose that heights of individuals in a population...

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Suppose that heights of individuals in a population follow the Normal(μ,σ2) distribu-tion. Suppose also that a random sample ofn= 25 individuals from this populationyielded sample mean ̄x= 155 centimeters and sample standard deviations= 10centimeters. Assume the following prior specification for (μ,σ2).μ|σ2∼N(160,σ2/10)σ2∼Inv-χ2(4,15).[Using the same notations as in Lecture Note 6.](a) Find the conditional posterior distribution ofμgiven (σ2,x1,...,xn).(b) Find the marginal posterior distribution ofμ.(c) Find the 95% highest posterior density (HPD) credible region forμ.(d) Find a 95% posterior credible interval forσ2.(e) If a new individual is randomly selected from the population, what is the posteriorprobability that this person's height will be between 150 and 160 centimeters ?