## this is a fake data set

 X <- floor(seq(1,10.8,0.2))
 
 X2 <- X^2
 Y <- X2 - 10*X + rnorm(length(X))*X        
      # the "true" slope = 1, intercept = 0
      # sigma is proportional to X 


 lm1 <- lm(Y ~ X + X2) 
 summary(lm1) 
 plot(X,Y) 
   lines(X,lm1$fit, col="red", lwd=2)

 lm2 <- lm(Y ~ X + X2, weights = 1/X2)          # weight proportional to inverse variance
 summary(lm2)
   lines(X,lm2$fit, col="blue", lwd=2)

 plot(X, lm1$resid)