It is believed that Tai Chi, an ancient Chinese practice of exercice and meditatio, is more effective on relieving sumptoms of chronic painful fibromyalgia than a widely-used wellness education method. A study is designed to test wether this belief is true or not. The study randomly assigned 35 fibromyalgia patients to take a 12-week Tai Chi class, and another 35 fibromyalgia patients to take a wellness education class. The result of the study are as follows:

27 patients in the Tai Chi class felt better at the end of the class;

15 patients in the wellness education class felt better at the end of the class.

Let p₁and p₂denote the proportions of patients that will feel better by taking Tai Chi and wellness education classes, respectively, out of all fibromyalgia patients. The significance level for this test is set to a=0,01.

(a) Using p₁ and p₂, write the null and alternative hypotheses.

(b) Test this using p-value approach.

(c) Test this using the critical value approach.

(d) Do the conclusions you made in (b) and (c) agree ?