Bayesian Statistics - Mendel's Experiment with Peas.

Johann Gregor Mendel (1822-1884) studied theinheritance of seven different features in peas, including height, flower color, seed color, andseed 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 andobserved how the traits were inherited.For the height trait, Mendel's model suggests that 3/4 of the plants grown from a crossbetween tall and short height strains of pea lines will be of the tall height variety. Afterbreeding 1064 of these plants, 787 resulted as the tall height variety. The reasonable modelfor the number of tall height results from n experiments is binomial Bin(n,p). Complete a 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: p ? 3/4? (Hint - After finding posterior?(p|x), use the posterior distribution to find?from 0 to 3/4 of ?(p|x)dp and use software to evaluate that integral.)

(c) Find a 95% equitailed credible set for the true proportion of tall height plants obtained from the given cross. In other words, find the equi-tailed proportion for posterior?

2. Mendel's Experiment with Peas. Johann Gregor Mendel (1822-1884) studied the inheritance of seven different features in peas, including height, ower color, seed color, and seed shape. To do so, he rst 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 11 experiments is binomial 13mm, p). Complete a Bayesian model with beta 33(12, 4) prior on the unknown proportion p. (a) What are prior and posterior means? (b) Find posterior probability of hypothesis Ho : p S 3/ 4? (c) Find a 95% equitailed credible set for the true proportion of tall height plants obtained from the given cross