STAT301 TA : Jie Zhang / Joanna jiezhang@stat.wisc.edu DISCUSSION 11 1 Summary 1.1 Small samples from two populations (Independent random samples) ? A second look at small samples from two population ? When both populations have the same standard deviation ?1 = ?2 = ?. We use S2pooled = (n1?1)S21+(n2?1)S22 n1+n2?2 , df = n1 +n2 ?2.Rule: When 1/2 ? s 1 s2 ? 2, we assume ?1 = ?2 = ?, and use Spooled. ? When both populations have the different standard deviations ?1 negationslash= ?2. We do NOT use Spooled. ? Sample samples CI for µ1 ?µ2. A 100(1??)% confidence interval for µ1 ?µ2 is given by ¯X ? ¯Y ±t?/2 radicalBigg S21 n1 + S22 n2 where df= smaller of n1 ?1 and n2 ?1. ? Testing H0 : µ1 ?µ2 = ?0 with small samples. Test statistic is T = ¯X ? ¯Y ??0 radicalBig S21 n1 + S22 n2 , df = smaller of n1 ?1 and n2 ?1 . Level ? Rejection Region is ? R : T ? t? when H1 : µ1 ?µ2 > ?0. ? R : T ? ?t? when H1 : µ1 ?µ2 < ?0. ? R : |T| ? t?/2 when H1 : µ1 ?µ2 negationslash= ?0. 1.2 Small samples inferences about the Mean Difference ? (Matched pairs samples) Assumethat thedifferences, Di = Xi?Yi are arandom samplefrom anN(?,?D). Let ¯D ? A 100(1??)% CI for ? is given by ¯D±t?/2 SD? n where t?/2 is based on df = n?1. ? A test of H0 : ? = ?0 is based on the test statistic T = ¯D??0 SD/?n, df = n?1 Office:1335N MSC 1 Office Hour: W 1:00-3:00pm, 4185 MSC
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