Scores for men on the verbal portion of the SAT-1 test are normally distributed

Question

Scores for men on the verbal portion of the SAT-1 test are normally distributed with a mean of 509 and a standard deviation of 112. (a) If 1 man is randomly selected, find the probability that his score is at least 571.5. (b) If 19 men are randomly selected, find the probability that their mean score is at least 571.5. 19 randomly selected men were given a review course before taking the SAT test. If their mean score is 571.5, is there a strong evidence to support the claim that the course is actually effective? (Enter YES or NO)

Explanation / Answer

a_P(X>571.5)=1-P(X<571.5)=1-P(Z<(571.5-509)/112)=1-P(Z<0.5580)=1-0.7116=0.2884

b)for std error =std deviation/(n)1/2 =25.6946

_P(X>571.5)=1-P(X<571.5)=1-P(Z<(571.5-509)/25.6946)=1-P(Z<2.4324)=1-0.9925=0.0075

c)Yes as probabilty of that is very less

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