Perform a designed-based permutation test to decide whether the response of treatment A for each patient is the same as the response of treatment B for each patient. Assume that a block of 4 patients will receive one of the six treatment assignments AABB, ABAB, ABBA, BAAB, BABA, and BBAA with equal probability. The response values of the 4 patients are y1 = 21, y2 = 18, y3 = 11 and y4 = 31. Patients 1 and 4 received treatment A, and patients 2 and 3 received treatment B.

(a) Please state the null hypothesis and alternative hypothesis.

(b) Please use a statistics test to tell if the response of treatment A is significantly different from the response of treatment B at the 0.05 level. Please provide the manual calculation.