The Rockwell hardness of a metal is determined byimpressing a hardened point into the surface of themetal and then measuring the depth of penetration of thepoint. Suppose the Rockwell hardness of a particularalloy is normally distributed with mean 70 and standarddeviation 3.a. If a specimen is acceptable only if its hardness isbetween 67 and 75, what is the probability that a randomlychosen specimen has an acceptable hardness?b. If the acceptable range of hardness is (70 2 c, 70 1 c),for what value of c would 95% of all specimens haveacceptable hardness?c. If the acceptable range is as in part (a) and the hardnessof each of ten randomly selected specimens isindepen-dently determined, what is the expectednumber of acceptable specimens among the ten?d. What is the probability that at most eight of ten independentlyselected specimens have a hardness of lessthan 73.84? [Hint: Y 5 the number among the tenspecimens with hardness less than 73.84 is a binomialvariable; what is p?]
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
deviation 3.
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 2 c, 70 1 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
indepen-dently determined, what is the expected
number of acceptable specimens among the ten?
d. What is the probability that at most eight of ten independently
selected specimens have a hardness of less
than 73.84? [Hint: Y 5 the number among the ten
specimens with hardness less than 73.84 is a binomial
variable; what is p?]
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