Suppose X and Y are the scores that a student will receive, respectively, on the verbal...

Suppose X and Y are the scores that a student will receive, respectively, on the verbal and math portions of the SAT test. Further suppose that X and Y are both Nor(700, 10000) and that Cov(X, Y ) = 2500. Find the probability that the total score, X + Y , will exceed 1500. (You can assume that X + Y is normal.)

Solutions

Expert Solution

W: Total Score

W=X+Y

E(W) = E(X+Y) = E(X)+E(Y)= 700+700=1400

Var(W) = Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y) = 10000+10000+2500=22500

Therefore W follows normal distribution with mean 1400 and variance 22500 or Standard deviation: = 150

probability that the total score, X + Y , will exceed 1500 = P((X+Y) > 1500) = P(W>1500)

P(W>1500) = 1-P(W1500)

Z-score for 1500 = (1500-mean)/Standard deviation = (1500-1400)/150 = 100/150=0.67

From standard normal tables : P(Z0.67)=0.7486

P(W1500)=P(Z0.67) = 0.7486

P(W>1500) = 1-P(W1500) = 1-0.7486= 0.2514

Probability that the total score, X + Y , will exceed 1500 = P((X+Y) > 1500) = P(W>1500) = 0.2514

orchestra answered 1 year ago

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