Key to assignment 1 

1.(a) >treat<-factor(treat) 
      >summary(aov(imp~treat))
      
      # At least 2 treatment means are different.
      

  (b) part 1:

      >D1<-ifelse(treat=="A", 1,0)
      >D2<-ifelse(treat=="B", 1,0)
      >summary(lm(imp~D1+D2))

      # Comment on A vs C (A-mean is significantly smaller than C-mean)
      # Comment on B vs C (B-mean is NOT significantly smaller than C-mean) 
      # C and B are better than A

      part 2:

      >treat<-factor(treat)
      >summary(lm(imp~treat))
      
      # Comment on A vs B (B-mean is significantly larger than A-mean)
      # Comment on A vs C (C-mean is significantly larger than A-mean) 
     
     

2.(a) >plot(x1, y)
      >plot(x2, y) 

  (b) >cor(x1, y)
      >cor(x2, y)

      x1 more correlated with y. Select x1.

      >reg1<-lm(y~x1)
      >summary(reg1)
      
      >qqnorm(reg1$residuals)
      >qqline(reg1$residuals) # normality holds 
  
      >plot(reg1$fitted, reg1$residuals) # constant variance 

      Regression coefficient 3.49 (from summary). p-value 5.26e-08 (almost 0).
      Significant.

  (c) >reg2<-lm(y~x2)
      >summary(reg2) 
 
      Regression coefficient 1.83 (from summary). p-value 0.0374.
      Significant.

  (d) >reg12<-lm(y~x1+x2)
      >summary(reg12)
      
      R^2 is 0.9372.

      93.72% of variability in y is explained by x1 and x2 together.
     
      R^2(d) < R^2(b)+R^2(c)= 1.0337 (more than 100% !!!)
      Indicates some correlation between x1 and x2.