Mendel's Experiment with Peas.

Johann Gregor Mendel (1822-1884) studied the inheritance of seven different features in peas, including height, flower color, seed color, and seed shape. To do so, he first established pea lines with two different forms of a feature, such as tall vs. short height. He grew these lines for generations until they were pure-breeds (always produced offspring identical to the parent), then bred them to each other and observed how the traits were inherited. For the height trait, Mendel's model suggests that 3/4 of the plants grown from a cross between tall and short height strains of pea lines will be of the tall height variety. After breeding 1064 of these plants, 787 resulted as the tall height variety. The reasonable model for the number of tall height results from n experiments is binomial B in (n,p). Complete Bayesian model with beta Be (12,4) prior on the unknown proportion p.

(a) What are prior and posterior means?

(b) Find posterior probability of hypothesis H0:p3/4?

(c) Find a 95% equitailed credible set for the true proportion of tall height plants obtained from the given cross.