The average american adult has completed an average of 11.25 years of education with a standard deviation of 1.75 years. a random sample of 75 adults is taken.
a) what is the probability that the sample will have a mean more than 11.5?
b) what is the probability that the sample will have a mean between 11 and 12 years?

Let X be a random variable representing the mean number of years of education completed by average American adult.
a.) P(X > 11.5) = 1 - P(X ≤ 11.5) = 1 - P(z ≤ (11.5 - 11.25)/(1.75/sqrt(75)))= 1 - P(z ≤ 1.24) = 1 - 0.89199 = 0.108

b.) P(11 < X < 12) = P(X < 12) - P(X < 11) = P(z < (12 - 11.25)/(1.75/sqrt(75))) - P(z < (11 - 11.25)/(1.75/sqrt(75))) = P(z < 3.712) - P(z < -1.237) = P(z < 3.712) - [1 - P(z < 1.237)] = P(z < 3.712) + P(z < 1.237) - 1 = 0.9999 + 0.89199 - 1 = 0.89189