PLEASE ANSWER ALL PARTS! Consider a Stanford University class where a student’s
ID: 3176124 • Letter: P
Question
PLEASE ANSWER ALL PARTS!
Consider a Stanford University class where a student’s ultimate grade depends on the total score the student receives on 10 separate, weekly exams. The student’s final class score is found by adding up all 10 exams; each exam has a maximum score of 100 points.
Let T = the sum of a student’s 10 exam scores. That is, T is the student’s final class score (i.e., the total of all 10 exams added together). Notice T ranges from 0 up to 1000.
Based on past experience, the instructor has found that T is normally distributed with an expected value of 684 and a standard deviation of 48. The instructor assumes this distribution for T when grading and gives the top 19% of students an A grade.
(a) Determine the lowest total number of exam points (i.e., the lowest value of T) needed for a student to earn an A grade.
One student in the class, Amy, estimates that her performance on any individual exam is given by the distribution:
Exam score: 50 Probability: .15
Exam score:60 Probability: .20
Exam Score:70 Probability: .25
Exam Score:80 Probability:.25
Exam Score:90 Probability: .15
Assume each of her 10 exam scores is independent of any other exam score.
(b) Using a normal distribution approach, determine the probability Amy will receive an A grade in the course.
(c) Suppose that, after the first 8 exams, Amy has an average score of 71.50. Determine the probability she ultimately receives an A grade.
(d) Now assume the rules on receiving an A grade change. Specifically, the instructor no longer restricts A’s to the top 19%. Rather, an A grade is given to any student who averages at least 69.50 on all exams.
(i) Suppose the distribution on Amy’s individual exam score is the same as above but the number of exams change (i.e., it is no longer 10). In order to increase her probability of receiving an A grade, should Amy want more or fewer exams? Please explain your response, arguing from the viewpoint of z-values. (No explicit calculation need here; just logical reasoning.)
(ii) Does your response to part (i) change if the instructor gives A grades to students who must average at least 72 on all exams (as opposed to 69.50)? Please explain your reasoning.
(iii) Now assume 10 exams with A grades going to those students who average at least 69.50 on all exams. But, now suppose Amy’s exam scores are not independent. Rather, assume the correlation coefficient between any two exams is a constant value (). Is Amy’s probability of earning an A higher if > 0 or if < 0? Again, no explicit calculation is necessary here but please give a convincing argument in your response. You may continue to assume Amy’s total class score is normally distributed (i.e., the correlation between exams is not extreme enough to severely alter the normal distribution shape).
Explanation / Answer
mean = 684 and std. dev. = 48
(A)
For 0.19, z value is 0.878
Minimum number of exam points required to score A grade is,
xbar = mean + z*sigma
xbar = 684 + 0.878*48 = 726.144
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