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 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?]

Answer 1

a. The probability of a randomly chosen specimen having an acceptable hardness can be calculated by finding the area under the normal distribution curve between the specified hardness values. b. The value of c can be determined by finding the z-scores corresponding to the specified acceptable range and solving an equation. c. The expected number of acceptable specimens among ten can be calculated using the binomial distribution. d. The probability that at most eight out of ten specimens have a hardness of less than a given value can be calculated using the binomial distribution.

Step-by-step explanation:

a. To find the probability that a randomly chosen specimen has an acceptable hardness, we need to find the area under the normal distribution curve between the hardness values of 67 and 75. This can be calculated using the standard normal distribution table or a statistical software. The probability can be written as P(67 ≤ X ≤ 75), where X is the hardness of the specimen.

b. To find the value of c, we need to determine the hardness values that encompass 95% of all specimens. This can be calculated using the standard normal distribution table or a statistical software. We need to find the z-scores corresponding to the upper and lower bounds of the acceptable range and solve for c in the equation P(-c ≤ Z ≤ c) = 0.95, where Z is the standard normal random variable.

c. Since each specimen's hardness is independently determined and the probability of an acceptable hardness is the same for each specimen, we can model the number of acceptable specimens among the ten as a binomial distribution. The expected number of acceptable specimens can be calculated using the formula E(X) = n * p, where n is the number of trials (in this case, ten) and p is the probability of success (the probability of acceptable hardness).

d. To find the probability that at most eight of ten independently selected specimens have a hardness of less than 73.84, we can use the binomial distribution. The probability can be calculated by summing the probabilities of having 0, 1, 2, ..., 8 successes out of ten trials with the given probability of success. Alternatively, we can use a binomial cumulative distribution function (CDF) to find the probability.

Answer 2

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