It is believed that cross-fertilized plants produce taller offspring than self-fertilized plants. In order to obtain an estimate on the proportion y of cross-fertilized plants that are taller, an experimenter observes a random sample of 15 pairs of plants exactly the same age, with each pair grown in the same conditions with one cross-fertilized and the other self-fertilized. Based on previous experience, the experimenter believes that the following are possible values of p and prior probabilities for each value (prior weight), π(θ):

From the experiment, it is observed that in 13 of 15 pairs, the crossfertilized is taller
(a) Create a table with columns for prior, likelihood of θ given sample, prior times likelihood, and posterior probability of θ. Based on the posterior probabilities, what value of θ has the highest support? Also, find E(θ) based on the posterior probabilities.
(b) Redo part (a) with a completely non informative prior, that is, take the prior for the proportion θ as one of the equally spaced values 0, 0.1, 0.2, . . ., 0.9, 1. Also assign for each value of θ the same probability, π(θ) = 1 / 11.
(c) Calculate the MLE of θ and compare it with the Bayesian estimate.

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0.80 0.03 0.82 0.40 0.84 0.22 0.86 0.15 0.88 0.15 0.90 0.05 π(θ):