Lect24_L6_L7_handout-1

Lect24_L6_L7_handout-1 - lecture 24, Linear Regression II...

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Unformatted text preview: lecture 24, Linear Regression II Testing the Significance of the Slope Confidence and Prediction Intervals lecture 24, Linear Regression II Outline 1 Testing the Significance of the Slope 2 Confidence and Prediction Intervals lecture 24, Linear Regression II Testing the Significance of the Slope Confidence and Prediction Intervals lecture 24, Linear Regression II Review Simple linear regression model: y = β + β 1 x + ε, ε ∼ N ( ,σ 2 ) . Least squares point estimates b , b 1 for β and β 1 : b 1 = SS xy SS xx := ∑ n i = 1 ( x i- ¯ x )( y i- ¯ y ) ∑ n i = 1 ( x i- ¯ x ) 2 , b = ¯ y- b 1 ¯ x . Point estimate of σ : s = r SSE n- 2 = s ∑ n i = 1 ( y i- ˆ y i ) 2 n- 2 lecture 24, Linear Regression II Testing the Significance of the Slope Confidence and Prediction Intervals lecture 24, Linear Regression II Testing the Significance of the Slope Testing the Significance of the Slope A regression model is not likely to be useful unless there is a significant relationship between x and y To test the significance of a linear relationship, we use the null hypothesis: H : β 1 = versus the alternative hypothesis: H a : β 1 6 = . lecture 24, Linear Regression II Testing the Significance of the Slope Confidence and Prediction Intervals lecture 24, Linear Regression II Testing the Significance of the Slope Testing the Significance of the Slope ( ctd) b 1 = SS xy / SS xx is computed based on data from a sample as a point estimate of β 1 Can show that the population of all possible values of b 1 (for a given list of x values) is normal with mean β 1 and standard deviation σ b 1 = σ √ SS xx . • σ b 1 is unknown since σ is unknown, but can be estimated by s b 1 = s √ SS xx • Furthermore, it can be shown that ( b 1- β 1 ) / s b 1 has a t-distribution with degrees of freedom n- 2 . lecture 24, Linear Regression II Testing the Significance of the Slope Confidence and Prediction Intervals lecture 24, Linear Regression II Testing the Significance of the Slope Testing the Significance of the Slope ( ctd) Hence if the null H : β 1 = is true , then t = b 1 s b 1 = b 1 s / √ SS xx = b 1 p ( n- 2 ) SS xx √ SSE ! has a t-distribution with degrees of freedom n- 2. • And we have the following rules for testing : Null Alternative Reject H if H : β 1 = H a : β 1 6 = | t | > t α/ 2 , or, equivalently p (= Area under t outside of [-| t | , | t | ] ) < α H : β 1 ≤ H a : β 1 > t > t α , or, equivalently p (= Area under t to right of t ) < α H : β 1 ≥ H a : β 1 < t <- t α , or, equivalently p (= Area under t to left of t ) < α where t α , t α/ 2 and p are based on t with df = n- 2 . lecture 24, Linear Regression II Testing the Significance of the Slope Confidence and Prediction Intervals lecture 24, Linear Regression II Testing the Significance of the Slope Example: Fuel Consumption Case We want to test H : β 1 = versus H a : β 1 6 = ....
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Lect24_L6_L7_handout-1 - lecture 24, Linear Regression II...

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