Businesses actively recruit business students with well-developed higher-order cognitive skills (HOCS) such as problem identification, analytical reasoning, and content integration skills. Researchers conducted a study to see if improvement in students' HOCS was related to the students' GPA. (Data extracted from R. V. Bradley, C. S. Sankar, H. R. Clayton, V. W. Mbarika, and P. K. Raju, "A Study on the Impact of GPA on Perceived Improvement of Higher-Order Cognitive Skills," Decision Sciences Journal of Innovative Education, January 2007, 5(1), pp. 151-168.) The researchers conducted a study in which business students were taught using the case study method. Using data collected from 300 business students, the following quadratic regression equation was derived:
HOCS = -3.48 + 4.53(GPA) - 0.68(GPA)2
where the dependent variable HOCS measured the improvement in higher-order cognitive skills, with 1 being the lowest improvement in HOCS and 5 being the highest improvement in HOCS.
a. Construct a table of predicted HOCS, using GPA equal to 2.0, 2.1, 2.2,..., 4.0.
b. Plot the values in the table constructed in (a), with GPA on the horizontal axis and predicted HOCS on the vertical axis.
c. Discuss the curvilinear relationship between students' GPA and their predicted improvement in HOCS.
d. The researchers reported that the model had an r2 of 0.07 and an adjusted r2 of 0.06. What does this tell you about the scatter of individual HOCS scores around the curvilinear relationship plotted in (b) and discussed in (c)?