We will use the Minitab macro NVarICP, or the function nvaricp in R, to nd the posterior distribution of the standard deviation σ when we have a random sample of size n from a normal(μ, σ²) distribution and the mean μ is known. We have S x an inverse chi-squared(k) prior for the variance σ². This is the conjugate family for normal observations with known mean. Starting with one member of the family as the prior distribution, we will get another member of the family as the posterior distribution. The simple updating rules are

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where SST = S' SSST and ' = k+n, (y)2. Suppose we have five observations from a normal(u, 2) distribution where = 200 is known. They are: 206.4 197.4 212.7 208.5 203.4 (a) Suppose we start with a positive uniform prior for the standard devi- ation . What value of Sx an inverse chi-squared (k) will we use? (b) Find the posterior using the macro NVarICP in Minitab or the func- tion nvaricp in R. (c) Find the posterior mean and median. (d) Find a 95% Bayesian credible interval for .