# A Statistical Analysis of Grades

## Question

PROJECT

Consider the [new] regression problem,

Final = a + b Homework + c Midterm1 + d Midterm2 + Noise

- Find parameter estimates for a, b, c, and d.
- Predict the final-exam score of a student in this course who has received a total of 80% for homework, 75% for Midterm1, and 79% for Midterm2.
- Do the following tests (each of which will give a p-value):
A) compute RSS under the null and RSS and use them to form an F statistic and test H0: c=d
B) use the elliptic confidence region to test H0: c=d=0
C) use Bonferonni's rectangular confidence region to test H0: c=d=0
D) use Scheffe's rectangular confidence region to test c=d=0

Warning: You may not use a canned statistical package to compute the p-value.
Although you are encouraged to use a computer for your linear algebra computations, you must calculate the p-value yourself. If needed, develop the theoretical computations that are necessary in order to achieve this.
- What is the meaning of this test? Explain the meaning of this statistical test to someone, with at-best limited statistical knowledge, who wishes to take this entry-level course [whose data you now have] in the future.

REPORT:

Form teams of 2-3 people. Share the thinking. Divide the technical tasks. But:
- Write the report to your boss who runs "your statistical consulting firm."
"School term-paper" standards are generally not high enough.
- Needless to say, your report MUST be typed. If you are in, or plan to be in, the MSTAT program, then you might as well write your report in LaTeX-2e.
- Your report must have an introduction. You explain the background here.
- Your report must also have at least one technical section. This is where you explain your analysis in detail (and correct prose).
- Your findings are reported in a separate section.
- You may choose to write a "Conclusions" section, but this is entirely optional.
- If you use/cite technical material, then cite carefully a source and add the source in your bibilography. This remark applies to your textbook, anything that I have written, others' manuscripts and notes, etc.

## Solution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

1 Introduction
The purpose of this report is to statistically analyze to what extent midterm grades contribute equally to the ﬁnal grade. We base this work on a sample of grades provided to us from a ﬁrst year freshman course in statistics and consist of all the homework grades, all the midterm grades, and the ﬁnal exam grades of 32 students. We were in addition provided with the grading policy for this class. On the basis of these data we developed a linear regression model of the form Final = a + b × Homework + c × Midterm1 + d × Midterm2
with estimates of the coeﬃcients. We tested the null hypothesis that the weights of the midterms (’c’ and ’d’) are the same using four diﬀerent methods of statistical testing:
A: Appropriate F statistic.
B: Method of elliptical confidence regions.
C: Bonferonni's rectangular confidence region.
D: Scheffe's rectangular confidence region.
and report a p-value for each of these tests. For none of these tests was the p-value suﬃciently low (say under .05) to reject the null hypothesis.

2 Technical Section
2.1 Preparation and Normalization
We were provided with all the grades of 32 students in a ﬁrst year statistics class. For each student this consisted of 9 homework grades, 3 midterm grades, and a ﬁnal exam grade. We were also provided with the maximum grade possible for each homework, midterm, and the ﬁnal exam....
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