The level of saturated fats in a brand of cooking oil is normally distributed with mean μ = 15% and unknown variance σ². The percentages of saturated fat in a random sample of ten bottles of the cooking oil are:

(a) What is the equation for the shape of the likelihood function of the variance σ²?
(b) Use Jeffreys' prior distribution for the variance σ². Change the variable from the variance to the standard deviation to nd the prior distribution for the standard deviation σ.
(c) Find the posterior distribution of the variance σ².

(d) Change the variable from the variance to the standard deviation to find the posterior distribution of the standard deviation.
(e) Find a 95% Bayesian credible interval for the standard deviation σ.
(f) Test H₀ : σ ≤ .05 vs: H₁ : σ > .05 at the 5% level of significance.

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