The hardness of a metal is determined by impressing a hardened point to the surface of the metal and then measuring the depth of penetration (in micro meters) of the point. Suppose the hardness of a particular metal is normally distributed with mean 70 and standard deviation 3. (Assume that hardness is measured on a continuous scale). (a) If a specimen is acceptable when 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 was (70 – c, 70 + c), for what value of c would 95% of all specimens have acceptable hardness? (c) If the hardness of 10 randomly selected specimens is independently determined and the acceptable range is as same as the range in part (i), what is the expected number of acceptable specimens among 10? (d) What is the probability that at most 8 out of 10 independently selected specimens have a hardness of less than 73.84?
The hardness of a metal is determined by impressing a hardened point to the surface of the
metal and then measuring the depth of penetration (in micro meters) of the point. Suppose the
hardness of a particular metal is
(Assume that hardness is measured on a continuous scale).
(a) If a specimen is acceptable when its hardness is between 67 and 75, what is the probability
that a randomly chosen specimen has an acceptable hardness?
(b) If the acceptable
specimens have acceptable hardness?
(c) If the hardness of 10 randomly selected specimens is independently determined and the
acceptable range is as same as the range in part (i), what is the expected number of
acceptable specimens among 10?
(d) What is the probability that at most 8 out of 10 independently selected specimens have a
hardness of less than 73.84?
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