The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the
point. Suppose the Rockwell hardness of a particular alloy is normally distributed
with mean 70 and standard deviation 3. (Rockwell hardness is measured on a
continuous scale.)
a. If a specimen is acceptable only if its hardness is between 67 and 75, what is the
probability that a randomly chosen specimen has an acceptable hardness?
b. If the acceptable range of hardness is (70 - c, 70 + c) for what value of c would
95% of all specimens have acceptable hardness?
c. If the acceptable range is as in part (a) and the hardness of each of ten randomly
selected specimens is independently determined, what is the expected number of
acceptable specimens among the ten?