The diameters of Red Delicious apples in a certain orchard are normally distributed with a mean of 2.63 inches and a standard deviation of 0.25 inch.

a. What percentage of the apples in this orchard have diameters less than 2.25 inches?

b. What percentage of the apples in this orchard are larger than 2.56 inches in diameter?

A random sample of 100 apples is gathered, and the mean diameter obtained is x  2.56.

c. If another sample of size 100 is taken, what is the probability that its sample mean will be greater than 2.56 inches?

d. Why is the z-score used in answering parts a–c?

e. Why is the formula for z-score used in part c different from that used in parts a and b?