Consider two independent samples {x1, x2, , xn} from population N(1, 2 1 ) and {y1, y2, , yn} from population N(2, 2 2 ) respectively. We wish to test the following hypotheses:

H0 : 1 = 2 vs H1 : 1 > 2.

Assume 1 = 2 = = 1. Find the powers of the z-test (from introductory statistics course) in the following cases:

(i) the powers to detect a difference of 1 2 = 0.1 and sample sizes are n = 9, 25, 100 respectively;

(ii) the powers to detect a difference of 1 2 = 0.5 and sample sizes are n = 9, 25, 100 respectively;

(iii) the powers to detect a difference of 1 2 = 1.0 and sample sizes are n = 9, 25, 100 respectively;