A study published in the journalNeurology(May 22, 2001) examined whether the drug botulinum toxin A is helpful for recurring pain among patients who suffer from chronic low back pain. The 31 subjects who participated in the study were randomly assigned to one of two treatment groups: 16 received a placebo of normal saline and the other 15 received the drug itself. The subjects' pain levels were evaluated at the beginning of the study and again after eight weeks. The researchers found that 2 of the 16 subjects who received the saline experienced a substantial reduction in pain, compared to 9 of the 15 subjects who received the actual drug.

(a)Use these data to conduct a two-sided test of whether one-third of all patients would experience substantial pain reduction from a placebo. Report the hypotheses and p-value, along with a summary of your conclusion. Also provide specific details about how the p-value is calculated.

(b)Repeat (a) for testing whether one-tenth of all patients would experience substantial pain reduction from a placebo.