Consider two independent random samples. The first one Y1,1, , Y1,9 is a random sample from N (1, 2) and the second one Y2,1, , Y2,9 is a random sample from N (2, 2). The parameters 1, 2, 2 are unknown. We want to conduct hypothesis testing H0 : 1 = 2 v.s. H1 : 1 6 = 2 If we use the two-sample t-test, we would calculate the following test statistic T = Y1 Y2 S2 p ( 1 9 1 9 ) where Yi = 1 J J j=1 Yij , i = 1, 2 and S2 p = 2 i=1 9 j=1(Yij Yi)2 9 92 . If we use the F-test from one-way ANOVA, we would calculate the following test statistic F = SSB/(2 1) SSW/(2 (9 1)) where SSB = 9 2 i=1( Yi Y)2 and SSW = 2 i=1 9 j=1(Yij Yi)2. Show that F = T 2. (Hint: show that Y1 Y = 1 2 ( Y1 Y2) and Y2 Y = 1 2 ( Y1 Y2))

1. Consider two independent random samples. The first one Y1,1, . .. , Y1, is a random sample from N(u1, 02) and the second one Y2,1, . . . , Y2,9 is a random sample from N(M2, 02). The parameters M1, M2, " are unknown. We want to conduct hypothesis testing Ho : M1 = M2 V.S. H1 : M1 / /2 If we use the two-sample t-test, we would calculate the following test statistic T = Y1. - Y2. where Vi = }Ejl Yiji = 1,2 and $2 = Li=12j= (Yij-Yi.)2 9+9-2 . If we use the F-test from one-way ANOVA, we would calculate the following test statistic SSB/ (2 - 1) F = SSW/ (2 X (9 - 1)) where SSB = 92(Yi - Y..)2 and SSW = >21 _ _1 (Yij - Yi.)2. Show that F = T2. (Hint: show that Y1. - Y.. = }(Y1. - Y2.) and Y2. - Y. = -; (Y1. -Y2.))