From past experience, a professor knows that the test score of students taking a

Question

From past experience, a professor knows that the test score of students taking a final examination is a random variable with mean 65.

(a) Give an upper bound on the probability that a student’s test score will exceed 75.

(b) Suppose in addition the professor knows that the variance of a student’s test score is equal to 30. What can be said about the probability that a student will score between 55 and 75?

(c) How many students would have to take the examination so as to ensure, with probability at least 0.8, that the class average would be within 5 of 65?

Explanation / Answer

a)

P(x >=a) <= E(X)/a

P(x>-75) <= 65/75 = 13/15

upper bound on the probability that a student’s test score will exceed 75 = 13/15 = 0.86667

b)

P( |x-mean| >= k) <= Variance/k2

P (55 <= x <= 75) = P ( |x-65|<=10)

P ( |x-65|<=10) <= 30/100 = 0.30

upper bound on probability that a student will score between 55 and 75 is 0.30

c)

P{|average - mean| >= a } <= variance/(n*a2)

so, P (|average-65| >= 5) <= 30/(n*25)

and

P (|average-65| <5) >= 1 - 30/(n*25) = 0.80

30/(n*25) = 1-0.80 = 0.20

n = 30/(25*0.20) = 6

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