Millions of people suffer from migraine headaches. The costs in work days lost, medication, and treatment are measured in the billions of dollars. A study reported in the Journal of the American Medical Association (2005, 203: 2118–2125) described an experiment that examined whether acupuncture is an effective procedure in treating migraines. A random sample of 302 migraine patients was selected and divided into three groups.

Group 1 was treated with acupuncture; group 2 was treated with sham acupuncture (patients believed that they were being treated with acupuncture but were not); and group 3 was not treated at all. The number of headache days per month was recorded for each patient before the treatments began. The number of headache days per month after treatment was also measured.

a. Conduct a test to determine whether there are differences in the number of headache days before treatment between the three groups of patients.

b. Test to determine whether differences exist after treatment. If so, what are the differences?

c. Why was the test in part (a) conducted?