A school makes a test and its scores has average $22.8$ and standard deviation $4.8$. The distribution of scores is approximately normal.
a) Choose one person randomly. What is the probability that her score is at least 25?
b) Choose 25 persons randomly. What is the probability that the average of their scores is at least 25?
c) Which one of your computations gives more accurate probability?
I managed to do a) and b) but in c) I don't know the answer. I just assumed that the distribution of scores is exactly normal so exact probabilities are
So is the answer that computations has the same accuracy? Or how can I evaluate the error as the distribution is not exactly normal? Or if I evaluate the expressions by Sage:
sage: 1-N(1/sqrt(2*pi)*integrate(e^(-x^2/2),x,-infinity,11/24)) 0.323356489807950 sage: 1-N(1/sqrt(2*pi)*integrate(e^(-x^2/2),x,-infinity,55/24)) 0.0109624426491708
do I have to find out how accurate methods Sage uses to evaluate the integrals?