Reference no: EM133411624

Question: A physical education student is studying the effect of a simple training program on basketball free-throw shooting. She recruits a random sample of 100 college students to participate in her study. Each student first shoots 100 free-throws to establish a baseline success probability. Each student then takes 50 practice shots each day for a month. At the end of that time, each student takes 100 shots for a final assessment. Let Yi denote the ith student's difference between his/her accuracy rate (shots made/100) after the program and his/her accuracy rate before the program, where "the program" is taking 50 practice shots each day for a month. Assume that each Yi|θ∼iidN(θ,σ2), where θ is an unknown parameter and σ2 is a known number. Give two prior distributions for θ, explaining each choice in a sentence or two:

1.a diffuse (vague) prior

2. a subjective prior based on your knowledge of free throw shooting
Find a coin that you can use for this exercise. Assume that the number of heads that come up when your coin is flipped n
times is Bin(n, θ), where θ is the probability that your coin lands heads up when flipped. Assume that before flipping it your prior distribution for θ is Beta(4,4).

1.Plot the prior density function for θ in R.

2. Flip the coin one time, clearly report the outcome, and find the posterior distribution for θ conditional on the outcome of the flip, carefully showing your derivation for full points. Plot the posterior density function in R.

3. Now assume that the prior distribution for θ is the posterior distribution from part (b). Flip the coin again and report the new outcome. Find the the posterior of θ given the new coin flip and plot the posterior density function.

4. Assuming the original Beta(4,4) prior distribution for θ, find the posterior distribution of θ conditional on the outcome of both coin flips. Compare this posterior distribution to the one you obtained in part (c).