(i) Test for the existence of regression (the F-test). Carefully de ne the null and alternative hypothesis, and explain the result of any R output you obtain.

(ii) Which of the 12 explanatory variables are not signi cantly associated with quality given that the other 11 variables are included in the model?

(iii) Fit the model using the 11 covariates that are signi cantly associated with quality. Calculate the R² and adjusted-R² value for your new model, and for the full model that used all 12 covariates. You should have found that R² decreases upon removal of the unnecessary covariate, but that the adjusted-R2 increased. Explain this nding.

(iv) Carry out a partial F test to decide if inclusion of log(pH) provides a signi cant improvement over the model that just uses the 11 covariates. Repeat this analysis to decide whether adding (density) provides a signi cant improvement. Be sure to carefully de ne the null and alternative hypothesis and the procedure you use to do the test.