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# Suppose that heights of individuals in a population follow the Normall[p, oz] distribu tion. Suppose also that a random sample of n. = 25...

(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 ?

1. Suppose that heights of individuals in a population follow the Normall[p, oz] distribu—
tion. Suppose also that a random sample of n. = 25 individuals from this population
yielded sample mean it = 155 centimeters and sample standard deviation s = 1!]
centimeters. Assume the following prior speciﬁcation for flu, 0'2]. Mag N Nﬂﬁﬂgrﬂflﬂ)
0'2 N lnv-fﬂJﬁ}. [Using the same notations as in lecture Note 6.] {a} Find the conditional posterior distribution of p. given (0'2, 1:1 , . . . , In“).
{b} Find the marginal posterior distribution of pi. {c} Find the 95% highest posterior density (HPD) credible region for n.
(d) Find a 95% posterior credible interval for 0'”. {e} If a new individual is randomlyr selected from the population, what is the posterior
probabilityr that this person‘s height will be between 153 and 16!] centimeters T

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