Suppose your course grade depends on two test scores: X_1 and X_2. Each score is

Question

Suppose your course grade depends on two test scores: X_1 and X_2. Each score is a Gaussian (mu = 74, sigma = 16) random variable, independent of the other. (a) With equal weighting, grades are determined by Y = X_1/2 + X_2/2. An "A" grade requires Y greaterthanorequalto 90; what is P(A) = P(Y greaterthanorequalto 90)? Useful fact: the sum of two independent gaussian random variables is also a gaussian random variable. (b) A student proposes that only the better of the two exam scores M = max (X_1, X_2) should be used to determine the course grade. The professor agrees; what is P(A) = P(M greaterthanorequalto 90)? (c) In a class of 100 students, what is the expected increase in the number of A's awarded due to this change in policy?

Explanation / Answer

a) mean of two grades =74

and std deviation of sum =16/(2)1/2 =11.3137

hence P(Y>90)=1-P(Y<90)=1-P(Z<90-74)/11.3137)=1-P(Z<1.4142)=1-0.9214=0.0786

b)for one grade below 90 ; P(X<90)=P(Z<(16/16))=P(Z<1)=0.8413

hence probabilty that at least onegrade is more then 90 =1-P(no grade is more then 90)=1-(0.8413)2 =0.2921

c) expected number to increase =n(p2-p1) =100*(0.2921-0.0786)=21.35

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