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- Wisconsin
- University of Wisconsin - Madison
- Statistics
- Statistics 301
- Xiao
- Fall 2008 Semester Discussion Notes- #11

Patrick D.

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 σ1 = σ2 = σ. 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 σ1 = σ2 = σ, and use Spooled. • When both populations have the different standard deviations σ1 negationslash= σ2. We do NOT use Spooled. – Sample samples CI for µ1 −µ2. A 100(1−α)% confidence interval for µ1 −µ2 is given by ¯X − ¯Y ±tα/2 radicalBigg S21 n1 + S22 n2 where df= smaller of n1 −1 and n2 −1. – Testing H0 : µ1 −µ2 = δ0 with small samples. Test statistic is T = ¯X − ¯Y −δ0 radicalBig S21 n1 + S22 n2 , df = smaller of n1 −1 and n2 −1 . Level α Rejection Region is ∗ R : T ≥ tα when H1 : µ1 −µ2 > δ0. ∗ R : T ≤ −tα when H1 : µ1 −µ2 < δ0. ∗ R : |T| ≥ tα/2 when H1 : µ1 −µ2 negationslash= δ0. 1.2 Small samples inferences about the Mean Difference δ (Matched pairs samples) Assumethat thedifferences, Di = Xi−Yi are arandom samplefrom anN(δ,σD). Let ¯D • A 100(1−α)% CI for δ is given by ¯D±tα/2 SD√ n where tα/2 is based on df = n−1. • A test of H0 : δ = δ0 is based on the test statistic T = ¯D−δ0 SD/√n, df = n−1 Office:1335N MSC 1 Office Hour: W 1:00-3:00pm, 4185 MSC

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