For the data given in Exercise 6.5-3, use a t test to test H0: = 0 against H1: > 0 at the = 0.025 significance level.

Lecture 5 Nicole Rubenstein September 12, 2017 5 Class Notes (Section 2.10-2.13) De▯nition 5.1. Let S be a nonempty sample space. If B 1B ;2::;B N are events of S such that S n 1. i=1B i B [1B [ 2▯▯ [ B = n 2. Bi\ B j 8i 6 j then B 1B 2:::;B norm a partition of S. Theorem 5.1. Law of total probability. Let A be an event of nonempty sample space S, and 1 ;B2;:::;Bn be a partition of S. We have Xn P(A) = P(A \ B i: i=1 Proof. Since A is an event of S (i.e., A ▯ S), we have A = A \ S = A \ (B [ B [ ▯▯▯ [ B ) 1 2 n = (A \ B ) [ (A \ B ) [ ▯▯▯ [ (A \ B ): 1 2 n Note that B 1B 2:::;B ns a partition of S, so we have Bi\ B j for any i 6= j. Thus we conclude that A \ B i \ (A \ Bj) for any i6=j. Thus, we arrive at P(A) = P ((A \ B ) [ (A \ B ) [ ▯▯▯ [ (A \ B )) 1 2 n Xn = P(A \ B ): i i=1 Corollary 5.1.