Problem 4. (18 points) We want to test whether a medical treatment is useful in improving patient's condition. 25 patients are given the drug and 25 patients are given the placebo. Assume the level a = 0.05. Improvement No Improvement Treatment 23 2 Placebo 16 9 (a) (6 points) Use chi-square test to decide whether the patient's conditions de- pend on having the treatment. Compute the cut-off value of the rejection region and make a decision. (b) (6 points) Use the Fisher exact test to decide whether the treatment improves the patients' conditions. Compute the p-value and make a decision. Solve without using R. (c) (6 points) Denote p1 and 132 the probability of improvement in the treatment group and placebo group. We want to test: Hozp1=p2vs Hazp1>p2 Assume the status of each patient X,- as independent bernoulli experiment with X,- = 1 meaning improvement and Xi = 0 meaning no improvement. Use Normal approximation to develop a sensible approximate test for the above hypotheses. Compute the p-value and make a decision