5. You are studying a new medication for chronic headaches. For each patient, you measure the improvement in their headache symptoms on a scale from 0 to 10 after taking the new medication. In a...

5. You are studying a new medication for chronic headaches.  For each patient, you measure the improvement in their headache symptoms on a scale from 0 to 10 after taking the new medication.  In a sample of 30 random patients, the mean improvement was 5.6 points, and the standard deviation was 2.8 points.

Suppose the current best medication is known to give a mean improvement of 5 points for patients with chronic headaches.  Perform a hypothesis test to investigate whether the new medication gives a higher mean improvement than the current best medication.

6.a. Suppose the study in Problem 5 was actually a small "pilot" study of the new medication.  Now you are planning a larger study.  If the true mean improvement for patients taking the new medication is 0.5 points better than the current medication (so the effect size is 0.5), you want the hypothesis test from the large study to have a power of 0.95 (95% chance of correctly rejecting the null hypothesis)