Scores for men on the verbal portion of the SAT-I test are normally distributed

Question

Scores for men on the verbal portion of the SAT-I test are normally distributed with a mean of 509 and a standard deviation of 112. Note: Please round z-scores to 3 decimal places and give probabilities to at least 4 decimal places.

(a) If 1 man is randomly selected, find the probability that his score is at least 574.

(b) If 14 men are randomly selected, find the probability that their mean score is at least 574.

(c) 14 randomly selected men were given a review course before taking the SAT test. If their mean score is 574, is there a strong evidence to support the claim that the course is actually effective? (Enter YES or NO)

Explanation / Answer

mean = 509
sd = 112

a)
P(X >= 574)
= P(z > (574 - 509)/112)
= P(z > 0.5804)
= 0.2808

b)
P(X >= 574)
= P(z > (574 - 509)/(112/sqrt(14)))
= P(z > 2.1715)
= 0.0149

c)
Yes, the probability of getting score 574 in a sample of 14 is 0.0149 which is unusual.

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