Question

a migraine is a particularly painful type of headache, which patients sometimes wish to treat with acupuncture. to determine whether acupuncture relieves migraine pain, researchers conducted a randomized controlled study where 89 females diagnosed with migraine headaches were randomly assigned to one of two groups: treatment or control. 43 patients in the treatment group received acupuncture that is specifically designed to treat migraines. 46 patients in the control group received placebo acupuncture (needle insertion at non - acupoint locations). 24 hours after patients received acupuncture, they were asked if they were pain free. results are summarized in the contingency table below (allais, 2011). (round answers to 2 decimal places)

	pain free: yes	pain free: no	total
control	2	44	46
total	12	77	89

a) what percent of patients in the treatment group were pain free 24 hours after receiving acupuncture?
%
b) what percent of patients in the control group were pain free after 24 hours?
%
c) at first glance, does acupuncture appear to be an effective treatment for migraines? explain your reasoning.
no, because there were fewer people in the treatment group
yes, because a higher percentage of individuals in the treatment group were pain - free after 24 hours
yes, because more individuals in the treatment group were pain - free after 24 hours
no, because more than half of the individuals in the treatment group still experience pain 24 hours later
d) do the data provide convincing evidence that there is a real pain reduction for those patients in the treatment group? or do you think that the observed difference might just be due to chance?
yes, 23% is much higher than 4%
it is impossible to tell merely by comparing the sample proportions because the difference could be the result of random error in our sample

Explanation:

Step1: Calculate percentage for treatment group

The formula for percentage is $\frac{\text{Number of pain - free in treatment group}}{\text{Total in treatment group}}\times100$. There are 10 pain - free patients in the treatment group and 43 patients in total in the treatment group. So the percentage is $\frac{10}{43}\times100\approx23.26\%$.

Step2: Calculate percentage for control group

Using the same percentage formula, with 2 pain - free patients in the control group and 46 patients in total in the control group. The percentage is $\frac{2}{46}\times100\approx4.35\%$.

Step3: Evaluate effectiveness

Comparing the percentages, a higher percentage of individuals in the treatment group were pain - free after 24 hours. So acupuncture appears to be effective. The correct reasoning is that a higher percentage of individuals in the treatment group were pain - free after 24 hours.

Step4: Analyze evidence

Just comparing the sample proportions, we cannot be sure if the difference is real or due to chance. It is impossible to tell merely by comparing the sample proportions because the difference could be the result of random error in our sample.

Answer:

a) 23.26
b) 4.35
c) yes, because a higher percentage of individuals in the treatment group were pain - free after 24 hours
d) It is impossible to tell merely by comparing the sample proportions because the difference could be the result of random error in our sample