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Unformatted text preview: Review E ( b 1 ) = β 1 V ar ( b 1 ) = σ 2 S XX E ( b ) = β V ar ( b ) = σ 2 parenleftBig 1 n + ¯ x 2 S XX parenrightBig b 1 and b are maximum likelihood estimators (i.e. they are ˆ β 1 and ˆ β ) Lecture 3 – p. 1/33 Where we are We are covering up through Sec 2.6 in Myers today We will skip Sec 2.7 and go on to 2.8  2.10 next week Still in Faraway Chap 1 MATH 533: Covered so far up to Hocking 2.3, will cover 2.4  2.6 next week Lecture 3 – p. 2 / Statistical inference for β 1 Although we know the expectation and variance of our estimators, b * and b 1 , we also need distributions for the estimators to make inference Usually we make inference through hypothesis tests and by building confidence intervals In order to do this exactly, we’ll need to assume that the data are normally distributed, usually accomplished by assuming that ǫ i ∼ N (0 ,σ 2 ) Lecture 3 – p. 3/33 t − test for β 1 and β Can test using our estimator ˆ β 1 = b 1 and properties of it Remember ˆ β 1 = b 1 = S XY S XX = ∑ n i =1 y i ( x i − ¯ x ) ∑ n i =1 ( x i − ¯ x ) 2 The ǫ i are independent normals, so y i are independent normals The x i ’s are constant Therefore, ˆ β 1 is a weighted sum of normal random variables (and so is also normal) Lecture 3 – p. 4 / t − test for β 1 and β Know the expected value ( β 1 ) and variance ( σ 2 /S XX ) of ˆ β 1 , so ˆ β 1 ∼ Normal ( β 1 ,σ 2 /S XX ) Let SS Res = ∑ n i =1 ( y i − ˆ y i ) 2 = ∑ n i =1 ( y i − ( b * + ( x i − ¯ x ) b 1 )) 2 Can show that SS Res /σ 2 ∼ χ 2 n 2 (will prove in Chapter 3) Let s 2 = SS Res n − 2 Lecture 3 – p. 5/33 t − test for β 1 and β Basic statistical inference and results from before about SS Res yields under H : β 1 = 0 : t = b 1 σ/ √ S XX × σ s = b 1 s/ √ S XX ∼ t n 2 Can adjust to test H : β 1 = β 1 , t = b 1 − β 1 , s/ √ S XX ∼ t n 2 Lecture 3 – p. 6 / Inference for intercept Like ˆ β 1 , b = ˆ β is a linear combination of independent normals, ˆ β = ¯ y − ˆ β 1 ¯ x We also know the mean and the variance of ˆ β , so for H : β = β , we can use: t = b − β , s radicalbig (1 /n ) + (¯ x 2 /S XX ) ∼ t n 2 Lecture 3 – p. 7/33 Accounting data10 10 20 30 40 5 10 15 20 ACCOUNTINGRATE MARKETRATE Lecture 3 – p. 8 / Accounting data > attach(accountdata) # Fit linear model predicting MARKETRATE with ACCOUNTINGRATE > lm(MARKETRATE˜ACCOUNTINGRATE) Call: lm(formula = MARKETRATE ˜ ACCOUNTINGRATE) Coefficients: (Intercept)...
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 Spring '06
 STEELE
 Math, Normal Distribution, Regression Analysis, Null hypothesis, Statistical hypothesis testing, SSRES

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