1. An automobile piston manufacturing company has two plants in North America. Plant A produces pistons of which 2% are defective, while plant B produces about 10% defective pistons. An automobile manufacturer receives 60% of the pistons from plant A and 40% from plant B. They inspect the pistons before assembly. Answer the following questions. 1. Postulate Beta prior distributions for the defective rate for the two plants: plant A: p ~ Beta(do, Bo) and plant B: p ~ Beta(1, B1). Assume do + Bo = 1 + B1 = 100. 2. During inspection of 200 pistons by the automobile manufacturer, 15 pistons are found to be defective. Find the posterior distribution of the defective rate (p). 3. Find the 95% equi-tailed credible interval for p. 4. Find the 95% HPD credible set for p