A standard chemistry examination administered nationally by the American Chemical Society has a mean of 500 and a standard deviation of 90. What is the probability that the average of a random sample of examination scores of 25 students will be between 450 and 500?

Let X be a random sample representing the score of a student in the American Chemical Society Examination.

P (450 < x < 500) = P ((450 - 500) / (90/sqrt (25)) < z < (500 - 500) / (90/sqrt (25))) = P (-2.778 < z < 0) = P (z < 0) - P (z < - 2.778) = P (z < 0) - [1 - P (z < 2.778) ] = P (z < 0) + P (z < 2.778) - 1 = 0.5 + 0.99727 - 1 = 0.49727 = 49.7%