# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 12Errors in hypothesis testingMoo K. Chung[email protected]Oct 20, 2004Type I error:you rejectH0whenH0is true.P(Type I error) =P(rejectH0|H0true) =α.Theresultingαis called thesignificance levelof the testand the corresponding test is called alevelαtest.We will use test procedures that giveαless than aspecified level (0.05 or 0.01 usually).I believe that dogs are as smart as people. AssumeIQ of a dog followsXiN(μ,102). IQ of 10 dogsare measured: 30, 25, 70, 110, 40, 80, 50, 60, 100,60. We want to test if dogs are as smart as people bytestingH0:μ= 100vs.H1:μ <100.(1)Let¯Xbe a test statistic andR= (-∞,90)tobe a rejection region. Let’s compute the probabil-ity of making Type I error based on this testingprocedure. Under the assumptionH0is true,XiN(100,102).Under this condition,¯XN(100,10)andα=P(¯X <90).>help(pnorm)...Usage:... pnorm(q, mean=0, sd=1,...)...> pnorm(90,100,sqrt(10))[1] 0.0007827011By using this test procedure, it is highly unlikelyto make Type I error.(2)Let’s see what happens when we change the testprecedure. WhenR= (-∞,95)with the sametest statistic¯X,α=
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