In a 2006 study published in The New England Journal of Medicine, 78 pairs of patients with Parkinson's disease were randomly assigned to receive treatment (which consisted of deep-brain stimulation of a region of the brain affected by the disease) or control (which consisted of taking a prescription drug). The researchers found that in 50 of 78 pairs, the patients who received deep-brain stimulation had improved more than their partner in the control group. The parameter of interest is , the probability of doing better on treatment than control.

(a) Let Y denote the number of pairs in which the treatment patient did better than the control patient. What is the distribution of Y? Provide your null/alternative hypothesis. Be specific!

(b) Test the null hypothesis in part (a) using a type I error of 0.05.

(c) Construct a 95% confidence interval for and interpret it.

(d) In the paper, the authors claim that deep-brain stimulation is "more effective than medical Management". Based on your answers to (b) and (c), do you agree and why?

(e) Suppose you analyzed this data from a Bayesian perspective, using a Uniform[0, 1] prior distribution on . What is the posterior distribution of |y?

(f) Plot the posterior density f( | y).

(g) Calculate a 95% posterior credible interval for . If you use any program, make sure to provide your codes.

(h) What is the posterior probability that > 0.5?

(i) Based on your analysis in (e)-(h), do you agree that deep-brain stimulation is "more effective than medical management".