MATH 376 -- Mathematical Statistics
   First Multiple Regression Examples 
   April 16, 2012

   A)  Fitting a model  Y = beta_0 + beta_1 x_1 + beta_2 x_2 + epsilon
       to the data  (0,0,4), (1,2,3), (1,0,5), (0,1,7)
       (geometrically, this is finding a plane coming closest to
       those four data points in R^3)

   > x1 <- c(0,1,1,0)
   > x2 <- c(0,2,0,1)
   > y <- c(4,3,5,7)
   > firstex <- lm(y ~ x1 + x2)
   > summary(firstex)

   Call:
   lm(formula = y ~ x1 + x2)

   Residuals:
    1    2    3    4 
   -1.6 -0.8  0.8  1.6 

   Coefficients:

                Estimate Std. Error t value Pr(>|t|)
   (Intercept)    5.600      1.960   2.858    0.214
   x1            -1.400      2.653  -0.528    0.691
   x2            -0.200      1.600  -0.125    0.921

   Residual standard error: 2.53 on 1 degrees of freedom
   Multiple R-squared: 0.2686,     Adjusted R-squared: -1.194 
   F-statistic: 0.1836 on 2 and 1 DF,  p-value: 0.8552 

   B)  Fitting a model  Y = beta_0 + beta_1 x + beta_2 x^2 + epsilon
       to the data (0,1), (1,4), (2,7), (3,8)

   
   > x <- c(0,1,2,3)
   > y <- c(1,4,7,8)
   > secondex <- lm(y ~ x + I(x^2))
   > summary(secondex)

   Call:
   lm(formula = y ~ x + I(x^2))

   Residuals:
      1    2    3    4 
     0.1 -0.3  0.3 -0.1 

   Coefficients:
                Estimate Std. Error t value Pr(>|t|)
   (Intercept)   0.9000     0.4359   2.065    0.287
   x             3.9000     0.7000   5.571    0.113
   I(x^2)       -0.5000     0.2236  -2.236    0.268

   Residual standard error: 0.4472 on 1 degrees of freedom
   Multiple R-squared: 0.9933,     Adjusted R-squared:  0.98 
   F-statistic:  74.5 on 2 and 1 DF,  p-value: 0.08165