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 71 and standard deviation 3.

What is the probability that at most eight of ten independently selected specimens have a hardness of less than 74.84? [Hint: Y = the number among the ten specimens with hardness less than 74.84 is a binomial variable; what is p?]

Explanation / Answer

Mean = 71

Standard deviation = 3

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

P(X < 74.84) = P(Z < (74.84 - 71)/3)

= P(Z < 1.28)

= 0.8997

P(at most 8 of 10 specimens have a hardness of less than 74.84) = 1 - P(9) - P(10)

= 1 - 10C9 x 0.89979x(1-0.8997) - 0.899710

= 1 - 0.3874 - 0.3475

= 0.2651

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