slide13 - Hypothesis tests in linear regression Slope ^ ^ 1...

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1 Hypothesis tests in linear regression Slope: H 0 : β 1 = β 1,0 H 1 : β 1 ≠ β 1,0 Reject H 0 : β 1 = β 1,0 if |t 0 |>t α /2,n-2 Intercept: H 0 : β 0 = β 0,0 H 1 : β 0 ≠ β 0,0 Reject H 0 : β 1 = β 1,0 if |t 0 |>t α /2,n-2 ) ˆ ( se ˆ S / ˆ ˆ T 1 0 , 1 1 xx 2 0 , 1 1 0 β β β σ β β = = ) ˆ ( se ˆ S x n 1 ˆ ˆ T 0 0 , 0 0 xx 2 2 0 , 0 0 0 β β β σ β β = + = 2 Hypothesis tests in linear regression Special case: H 0 : β 1 = 0 H 1 : β 1 0 Fail to reject H 0 : β 1 = 0 is equivalent to concluding that there is no linear relationship between x and Y.
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3 Example: Sales vs. Advertising Advertising (mil.$) Sales (mil.$) 1.2 120 1.6 130 1.8 140 2.2 150 2.6 130 3.1 140 3.4 120 3.6 150 4.0 140 4.2 130 80 90 100 110 120 130 140 150 160 0 1 2 3 4 5 4 Analysis of variance Test for significance of regression. Analysis of variance identity Symbolical equation = = = + = n 1 i 2 i i n 1 i 2 i n 1 i 2 i ) y ˆ y ( ) y y ˆ ( ) y y ( E R T SS SS SS + = squares of sum Error : SS squares of sum gression Re : SS squares of sum correct Total : SS E R T
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5 Analysis of variance Test statistics Reject H 0 : β 1 = 0, if f 0 >f α ,1,n-2 E R E R 0 MS MS ) 2 n /( SS 1 / SS F = = 6 Confidence intervals
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