Stat 312: Lecture 16Other two sample testsMoo K. Chung[email protected]March 25, 2003Concepts1. Paireddata:foragivenpairedsample(X1, Y1),· · ·,(Xn, Yn)withXi=μXandYi=μY, a test statistic for testingH0:μX=μYcanbe based on one sample test.2. LetXi∼Bernoulli(pX)andYj∼Bernoulli(pY). LetˆpX=∑ni=1Xi/nandˆpY=∑mj=1Yj/m.Var(ˆpX-ˆpY) =pX(1-pX)n+pY(1-pY)m.3. Difference between population proportions: for largenandm, use aZ-statistic for testingH0:pX=pY:Z=ˆpX-ˆpY-(ˆpX-ˆpY)pVar(ˆpX-ˆpY)∼N(0,1).SincepXandpYare unknown, we estimate them fromthe samples.In-class problemsExample 1.10 students took two midterm exams.Studentk01 02 03 04 05 06 07 08 09 10Midterm 1k80 75 60 90 99 60 55 85 65 70Midterm 2k70 60 70 72 95 66 60 80 70 60Is the first exam easier than the second exam? Test it at level0.05.Solution.LetXiandYibe the first and the second midtermscores for thei-th student. Perform thet-test onH0:μY-μX= 0
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