(Solution) - In Exercise 7 23 the posterior distribution of 2 the variance -(2025 Original AI-Free Solution)

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In Exercise 7.23 the posterior distribution of ?2, the variance of a normal population, given S2, the sample variance based on a sample of size n, was found using a conjugate prior for ?2 (the inverted gamma pdf with parameters a and 3). Based on observing S2, a decision about the hypotheses Ho: ? ? 1 versus H0: ? > 1 is to be made.
(a) Find the region of the sample space for which P(a < l|s2) > P(a > l|s2), the region for which a Bayes test will decide that a ? 1.
(b) Compare the region in part (a) with the acceptance region of an LRT. Is there any choice of prior parameters for which the regions agree?