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1. [16 mark(s)] Please use R to do this question, and include your code along with your answers. An olympic diver is practicing a difcult dive. A year ago, the diver achieved a 70% success rate. To measure his ability this year, called 6, he performs N = 20 attempts and counts the number of successes 1:. From the structure of the experiment, we know the sampling distribution is $[6 ~ Binomial (N, 6). (1) Consider the following hypotheses about 6: s0 : a = 0.7 (2) H1 : a 72 0.7 (3) (a) [2 mark(s)] Use R to construct a vector of possible 6 values from 0 to 1 in steps of 0.01. (b) [2 mark(s)] Construct a vector of prior probabilities that assigns 50% prior probability to H0, and spreads the other 50% uniformly among the other possibilities. (c) [4 mark(s)] The observed number of successes is a: = 18. Calculate the posterior distribution for 6, and hence the posterior probability of Ho. (d) [2 mark(s)] The diver wants to use this analysis to predict whether he will succeed in doing this dive when he attempts it in a competition the next day. However, he doesn't know whether his success probability will be affected by nerves Let Mo be the hypothesis that nerves don't matter, so his success rate of 6 (measured while practicing) still applies in the competition. Calculate the posterior predictive probability of success for the dive in the competition, given MD. (e) [2 mark(s)] Consider another hypothesis M1, which states that if the diver's success rate is 6 in practice sessions, it will be 62 in the competition, due to nervesl. Calculate the predictive probability of success on the dive in the competition, given M1. (1') [4 mark(s)] In the competition, the dive is a failure. Calculate the Bayes Factor for M1 over M0, and the posterior odds ratio for M1 over M0, assuming a prior odds ratio of 1