Problem 2: Normals and Making the Curve

Due to your excellent work to date, you've been hired as a Course Assistant to CSCI3022! The new instructor isn't really in the mood to write new problems, so they've turned to you to create the exam and ensure that it has a reasonable distribution of grades.

For each of the prompts below, you should at the very least set up and justify your calculation by hand (in Markdown/Mathjax), before executing any calculations in Python. After that, feel free to use python to actually do the calculations and get decimal results. You should use absolutely use Python to retrieve critical values of a standard normal random variable (using a Z chart is so old school!).

Part A: Suppose that you've created an exam with a mean of 75 points and a standard deviation of 15 points. These scores are normally distributed Thanks to including some extra credit, note that there's some possibility of scores above 100... which is fine, because those students are (fire emoji).

The administration has told you that they'd like to track your "B/C" students, and wants data regarding any student whose exam grade fell between 78 and 84. What is the probability that an individual students' (say, the ith) grade Xi falls between 78 and 84?

Part B: Suppose your class has 30 students. What is the probability that the overall average for the exam X falls between 78 and 84?

Part C: Suppose you had control over enrollment, and could choose the number of students n in your class. What value would you choose for n to maximize the probability that the overall average for the exam X falls between 78 and 84?

Hint: you can brute force it using a similar calculation to that in B!

Part D: You've realized that no matter what you do, the standard deviation for exam scores basically never deviates from 15 points. But you can affect the mean by writing slightly easier/tougher problems or providing better hints along the way! For another class of 30 students, what value would you have to change the exam average to be to ensure that there is a 97% chance that the class average score is above 78?