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.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 (69-C, 69+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?

d. What is the probability that at most eight of ten independently selected specimens have a hardness of less than 72.84

b. If the acceptable range of hardness is (69-C, 69+C), for what value of C would 95% of all specimens have acceptable hardness?

c.

d. What is the probability that at most eight of ten independently selected specimens have a hardness of less than 72.84

a) P(67 < x < 75) = 0.81648

b)

c) 8.1648

d)

Step-by-step explanation:

a) probability that a randomly chosen specimen has an acceptable hardness within 67 and 75.

This is a normal distribution problem with

Mean = μ = 71

Standard deviation = σ = 3

We first standardize 67 and 75

For 67,

z = (x - μ)/σ = (67 - 71)/3 = - 1.33

For 75

z = (x - μ)/σ = (75 - 71)/3 = 1.33

P(67 < x < 75) = P(-1.33 < z < 1.33)

= P(z < 1.33) - P(z < -1.33)

= 0.90824 - 0.09176

= 0.81648.

b) P(69-C < x < 69+C) = 0.95

Let the z-score of 69-C

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?

E(X) = np

n = sample size = 10

p = proportion of the sample with acceptable hardness as calculated in (a) = 0.81648

E(X) = 10 × 0.81648 = 8.1648

d) the probability that at most eight of ten independently selected specimens have a hardness of less than 72.84

Firat of, we calculate the probability that one

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step-by-step explanation: