Let a random sample of 5 observations from a normal (, 2 ) distribution (where it

Let a random sample of 5 observations from a normal (μ, σ²) distribution (where it is known that the mean μ = 25) be

(a) What is the equation for the shape of the likelihood function of the variance σ²?
(b) We believe (before looking at the data) that the standard deviation is as likely to be above 4 as it is to be below 4. (Our prior belief is that the distribution of the standard deviation has median 4.) Find the inverse chi-squared prior with 1 degree of freedom that fits our prior belief about the median.
(c) Change the variable from the variance to the standard deviation to find the prior distribution for the standard deviation σ.
(d) Find the posterior distribution of the variance σ².
(e) Change the variable from the variance to the standard deviation to find the posterior distribution of the standard deviation.

(f) Find a 95% Bayesian credible interval for the standard deviation σ.

(g) Test H₀ : σ ≤ 5 vs: H₁ : σ > 5 at the 5% level of significance.

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