While studying, I came across a question regarding the critical value for the Pearson chi-square test (pg. 1.66) and the LRT (pg. 1.68) for the hypothesis test, testing H0: pi1 – pi2 = 0 vs Ha: pi1 – pi2 ne 0. The Pearson chi-square critical value is chi-square (1, 1 – alpha) while the LRT critical value is chi-square (1, 1 – alpha/2). Why does it appear the Pearson is conducting a one sided test while the LRT appears to be conducting a two sided test?