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Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 20 Inference on Intercept Moo K. Chung [email protected] April 8, 2003 Concepts 1. Least-squares estimator ˆ β 0 = ¯ Y - ˆ β 1 ¯ x is distributed as normal with mean β 0 and variance σ 2 x 2 /S xx . 2. Testing H 0 : β 0 = 0 is based on T = ˆ β 0 - β 0 S ˆ β 0 t n - 2 , where S ˆ β 0 = σ 2 x 2 /S xx . 3. Coefficient of determination measures the proportion of variability explained by the fitted model and it is defined as r 2 = 1 - SSE SST 4. Since SST = S yy and SSE = S yy - S 2 xy /S xx , we can see that r 2 = S 2 xy / ( S xx S yy ) . r is called the sample correlation coefficient and it will be studied in Lecture 21. In-class problems Example 1. Hooke’s law states that the length change y in spring is proportional to applied force x , i.e. y = β 1 x x (kg) 29.4 39.2 49.0 58.8 68.6 78.4 y (mm) 4.25 5.25 6.50 7.85 8.75 10.10 Fit the model Y = β 0 + β 1 x + and see if β 0 is in fact 0. > ra<-cov(y,y)/cov(x,x)- (cov(x,y)/cov(x,x))ˆ2 > S0<-sqrt(mean(xˆ2)/4*ra) > b0<-mean(y)-mean(x)*cov(x,y)/cov(x,x) > b0/S0 [1] 4.013 > 2*pt(-4.013,4) [1] 0.016 >x<-c(29.4,39.2,49.0,58.8,68.6,78.4) >y<-c(4.25,5.25,6.50,7.85,8.75,10.10)
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Unformatted text preview: >summary(lm(y˜x)) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.66 0.16 4.00 0.016 x 0.12 0.003 41.24 2.07e-06 Multiple R-Squared: 0.9977 Example 2. Suppose that data ( x j , y j ) satisfy a circular rela-tionship Y j = q 10 2-x 2 j + ± j . This is a nonlinear relation-ship so we expect not to have a good linear fit. 30 40 50 60 70 80 5 6 7 8 9 10 x y Figure 1: Hooke’s law.-10-5 5 10 2 4 x Figure 2: Complete lack of linear fit. >x<--10:10 >y<-sqrt(10ˆ2 -xˆ2) >summary(lm(y˜x)) Coefficients: value Pr(>|t|) (Intercept) . ... 1.24e-09 x .... 4.47e-17 Multiple R-Squared: 1.378e-030 Self-study problems Construct 100(1-α )% confidence intervals for both β and β 1 . Show Var ( ˆ β ) = σ 2 x 2 /S xx . Example 12.9....
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