Question

The mean score of the SAT Exams at a particular college is 1830 with a standard deviation of 160.

a. What is the probability that a group of thirty-six students in the incoming class will have a mean score greater than 1900?

b. What is the probability that two groups of students, one with 40 students and one with 50 students will have mean scores that are different by more than 35?

Answer 1

a)for std error of mean =std deviation/(n)^1/2 =160/(36)^1/2 =26.6667

and as z score =(X-mean)/std deviation

therefore

probability that a group of thirty-six students in the incoming class will have a mean score greater than 1900

=P(X>1900)=1-P(X<1900)=1-P(Z<(1900-1830)/26.6667)=1-P(Z<2.625)=1-0.9957 =0.0043

b) let group with 40 students has mean score of X and that with 50 students has mean score of Y

therefore let difference is W.

therefore mean of W =E(X)-E(Y)=1830-1830=0

and std deviation of W =(160²/40+160²/50)^1/2 =33.9411

hence  probability that   mean scores that are different by more than 35 =P(|W|>35)=1-P(-35<W<35)

=1-P((-35-0)/33.9411<Z<(35-0)/33.9411)=1-P(-1.0312<Z<1.0312)=1-(0.8488-0.1512)=1-0.6976 =0.3024