1. Suppose you are given a dataset that consists of a random sample of tasters, on which the following variables were obtained:

(y) Zpref = taste preference for green beans stored in brine solution (data is centered)

(x1) saltc   = level of salt in brine

(x2) sugarc  = level of sugar in brine

(x3) saltc2  = salt*salt  or salt-squared

(x4) sugarc2 = sugar*sugar or sugar-squared

The researchers are interested in how varying the levels of salt and/or sugar in a brine solution influence taste preferences of canned green beans packed in a brine solution.  Please answer the following questions. 

Notation:  You may write models without subscripts

                         (e.g. y = b0 + b1 x1 + e    is fine)

a. Write the full and restricted models which - in a models comparison framework - would evaluate the null hypothesis that neither salt nor sugar has a significant linear or curvilinear relationship with taste preference.

b. Using the full model from part a, write the restricted model that implies that sugar has no influence on taste preference. 

c. Suppose you were to see the following SAS code in your program editor window:

   PROC REG;

   MODEL zpref = saltc sugarc saltc2 sugarc2;

   QTERMS: Test saltc2=0, sugarc2=0;

c1) In WORDS, what is the hypothesis being tested in the test statement labeled QTERMS? 

c2) Write the full and restricted models used to evaluate the QTERMS hypothesis.

d. Compared to the full model in a, write the restricted model that simultaneously implies that

1) The linear effect of sugar conditional on level of salt is zero, AND

2) The linear effects of salt and sugar are equal to each other, AND

3) The curvilinear effects of salt and sugar are equal to each other.  Give the SAS PROC REG code needed to evaluate this hypothesis.