15.1 The strength of an item is known to be normally distributed with mean 200 and unknown variance CJ’. A random sample of ten items is taken and their strength measured. The strengths are:

215 186 216 203 221 188 202 192 208 195

(a) What is the equation for the shape of the likelihood function of the variance CT’?

(b) Use a positive uniform prior distribution for the variance CT’. Change the variable from the variance to the standard deviation to find the prior distribution for the standard deviation CT.

(c) Find the posterior distribution of the variance CT’.

(d) Change the variable from the variance to the standard deviation to find

(e) Find a 95% Bayesian credible interval for the standard deviation CT.

(f) Test HO : D 5 8 vs. HI : 0 > 8 at the 5% level of significance.

the posterior distribution of the standard deviation.