Professor Moriarty has never taken a formal statistics course; however, he has heard about the bell-shaped curve and has some knowledge of the Empirical Rule for normal distributions. Professor Moriarty teaches as Honors Quantum Physics class in which he grades on the bell curve. He assigns letter grades to his students' tests by assuming a normal distribution and utilizing the Empirical Rule. The Professor reasons that if IQ and SAT scores follow a normal distribution, then his students' scores must do so also. Therefore, upon scoring the tests, he determines the mean and standard deviation for his class. He then uses the Empirical Rule to assign letter grades so that 68% of the students receive a "C," 95% receive "B-D," and 99.7% receive "A-F."
The following test grades occur on the midterm exam for his class: 78 85 93 62 82 76 74 73 91 66 89 88 86 94 65 90 84 92 94 92 82 85 80 77 52 84 78 83
a) You are working as Professor Moriarty's graduate assistant and he has asked that you use the Empirical Rule to determine which of these grades he should assign as "A," "B," "C," "D," and "F." After finding the mean and standard deviation for the midterm grades, give the interval of test scores that will qualify for each letter category. (In other words, what range of scores will earn an "A," "B," and so on?) Also give how many students will earn each letter grade using this grading scheme.
b) Determine the number of students who would receive an "A," "B," "C," "D," and "F" using a standard grading scheme where 90-100 earns an "A," 80-89 earns a "B," 70-79 earns a "C," 65-69 earns a "D," and below 65 earns an "F." Describe this grade distribution and contrast it with the one that results from using the bell curve.
c) As a student of statistics, you have some concerns about Professor Moriarty's use of the normal distribution in this context. Discuss your concerns with your classmates. Why might his practice not be statistically sound? Provide statistical evidence to support your position so that you can justify your argument against this method of grading. (Statistical evidence could include measures of relative position, charts or tables, explanations supported by statistical knowledge or analysis, etc.)