Suppose that Quantitative GRE scores are distributed normally with a mean of 150

Question

Suppose that Quantitative GRE scores are distributed normally with a mean of 150 and a standard deviation of 10.

(a) What fraction of students score above 155?

(b) Two students are chosen at random.

(i) What is the probability that the first has a score above 155?

(ii) What is the probability that the second has a score above 155?

(iii) What is the probability that both have a score above 155?
(iv) What is the probability that the sum of their scores is above 310?

(c) Suppose Verbal GRE scores are also distributed normally with a mean of 150 and a standard deviation of 10. Suppose that the correlation between an individual’s math and verbal scores is 0.80, and that scores are “jointly” normally distributed.

(i) Calculate the covariance between Quantitative and Verbal scores.

(ii) Find the mean and standard deviation of the sum of the Verbal and Quantitative scores.

(iii) What is the probability that the sum of the two scores is above 310?

(iv) A student tells you that her Quantitative GRE score is 160. What is the conditional mean and variance of her Verbal score. What is the probability that her Verbal score is above 155?

Explanation / Answer

a)
Z = (X - 150)/10
P(X > 155) = P(Z > ( 155 -150)/10)
= P(Z > 0.5)
= 0.3085

b)
(i) What is the probability that the first has a score above 155?
= 0.3085

(ii) What is the probability that the second has a score above 155?
=0.3085

(iii) What is the probability that both have a score above 155?
=0.3085^2 = 0.09517225
(iv) What is the probability that the sum of their scores is above 310?
Y = X1+X2 follow N(150+150 , 10^2+10^2)
P( Y > 310)
= P(Z >(310 - 300)/sqrt(200))
=P(Z > 0.707106)
= 0.2398

Please ask rest questions again

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