In an introductory statistics course, the following weights are applied to determine each student’s overall grade for the course.
Homework: 10%
Labs: 10%
Project 1: 5%
Final Project: 20%
Exam 1: 15%
Exam 2: 15%
Final Exam: 20%
Quiz: 5%
a. Use the assigned weightings to give an overall course score to each of the 32 students in the Grades data set. The top 15% of the students should get an A, the next 25% a B, the next 35% a C, the next 15% a D, and the final 10% an F.
b. Use the first principal component from a PCA of the Grades data instead of the assigned weights to create an overall score for each student in the class. What percentage of the total variability in grade components is explained by this score? Compare the weightings from the PCA to the instructor’s weightings. If the first principal component were used instead of the predefined weights, how many students would get a different grade in the course? Explain how changing the weightings influenced students’ final grades.
c. Repeat Part C, but do not standardize the data (use the covariance matrix instead of the correlation matrix). Explain why homework now has a much higher weight and is the primary variable in a student’s grade calculation. (Which variable has the highest variability in the unstandardized data set?)
d. What percentage of the variation in the data is explained by PC1 in Parts B and C? Do higher eigenvalues indicate a better analysis? Explain.