The Rockwell hardness of a metal is determined by impressing a hardened point in

Question

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. (a) If a specimen is acceptable only if its hardness is between 67 and 73, what is the probability that a randomly chosen specimen has an acceptable hardness? (Round your answer to four decimal places.) (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? (Round your answer to two decimal places.) ? the acceptable range is as in part a and the hardness of each of ten ando your answer to two decimal places.) y selected spec ens is independen y determine at s e expected number or acceptable spec mensa mon e en? ound specimens (d) What is the probability that at most eight of ten independently selected specimens have a hardness of less than 73.84? [Hint: Y- the number among the ten specimens with hardness less than 73.84 is a binomial variable; what is p?] (Round your answer to four decimal places.)

Explanation / Answer

a) P(X < A) = P(Z < (A - mean)/standard deviation

P(67 < X < 73) = P(X < 73) - P(X < 67)

= P(Z < (73-70)/3) - P(Z < (67 - 70)/3)

= P(Z < 1) - P(Z < -1)

= 0.8413 - 0.1587

= 0.6826

b) P(70-c < X 70+c) = 0.95

P(X < 70+c) = 0.95 + (1-0.95)/2

P(X < 70+c) = 0.975

(70+c - 70)/3 = 1.96

c = 5.88

c) Expected number of specimens among ten = 10x0.6826

= 6.83

d) P(X < 73.84) = P(Z < (73.84 - 70)/3)

= P(Z < 1.28)

= 0.8997

P(at most 8 specimens have hardness less than 73.84) = 1 - P(greater than 8 specimens have hardness less than 73.84)

= 1 - P(9) - P(10)

= 1 - 10C9x0.89979x(1-0.8997) - 0.899710

= 1 - 0.3874 - 0.3475

= 0.2651

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