Textbook page 467 offers a summary as well a two

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Unformatted text preview: and in Hypothesis Testing, the test statistic must be derived under H0. Textbook page 467 offers a summary as well. A two-sided (1–α)% C.I. for μX = σX2 = pX = ( μD = μX – μY = ( − ) ± ̂± ± ± ⁄( ⁄ ) ⁄ ⁄ ( − 1) ) , ̂ (1 − ̂ )⁄ ⁄ ⁄ ( ⁄( ) (known σX2) and ) ± ⁄ ( − 1) ⁄ (unknown σX2) ⁄ + where Di = Xi – Yi (known σX2 and σY2) and ( − )± pX – pY = ( ̂ − ̂ ) ± σX2/σY2 = ⁄ where “pooled” sample variance ⁄ ( − )± ⁄ ⁄ ( + + − 2) (unknown σX2 and σY2 but a “large” sample) + (unknown but equal σX2 and σY2) ( ) )( ( ) ) ⁄ μX = μ0 : z 0 = ( − ) ⁄ (known σX2) and t0 = ( − σX2 = σ02: χ02 = (nX – 1)SX2/σ02 ~ χ2(nX – 1) Test statistic for testing H0: pX = p0: z0 = ( ̂ − ) μD = μ0 : t 0 = ( − )⁄ ⁄ ( , ) × , ⁄ + = ( ( − 1, − 1) × )⁄ ⁄ ~ t(nX – 1) μX – μY = μ0 : z 0 = ( − z0 = ( − t0 = ( − pX – pY = 0 : z0 = ( ̂ − ̂ ) (unknown but equal σX2 and σY2) + − − − ) ) ) ~ t(n – 1) + + (known σX2 and σY2) and =( − ) (unknown σX2 and σY2 but “large” sample) + ~ t(nX + nY – 2) ∑ + where “pooled” σX2 = σY2: F0 = SX2/SY2 ~ F(nX – 1, nY – 1) = ∑ The rejection rule depends whether it is a one-sided or two-sided test. For simple linear regression =+ + = 1, ⋯ , with [ | ] = σ , the OLS estimators of β0 and β1 are ∀ , [ | ] = 0 and ∑ ( − )( − ) ∑ ( − ) = = ∑ (−) ∑ (−) Moreover, = and = = ; and − =∑ ( ) and distribution with the above means and variances, hence C.I.’s and hypothesis testing ’s are possible; given we estimate σ2 by =∑ ̂ ⁄( − 2). = +∑ ( ) . By CLT, these two estimators follow normal Final Remark: understand where the randomness comes from, i.e. which variables are random? Taylor 2 April 2009...
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