The Rockwell hardness (X) 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 an alloy is normally distributed with mean 69 and standard deviation 3. (Rockwell hardness is measured on a continuous scale.) Note: Take 10 decimal places after the ".", if the answer is a fraction, enter the fraction (a/b). Use a period (.) not a comma (,) for decimals.
a) If a specimen is acceptable only if its hardness is between 67 and 71, what is the probability that a randomly chosen specimen has an acceptable hardness?
P(67≤X≤71) = ?
b) If the acceptable range of hardness is (69−c,69+c), for what value of c would 96% of all specimens have acceptable hardness?
c = ?
c) If the acceptable range is as in part (a) and the hardness of each of 6 randomly selected specimens is independently determined, what is the probability that at most 4 of 6 independently selected specimens have a hardness of this acceptable range.
Y= the number of specimens have a hardness of this acceptable range.
P(Y≤4) = ?

Step-by-step explanation:

The Rockwell hardness (X) 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 an alloy is normally distributed with mean 69 and standard deviation 3. (Rockwell hardness is measured on a continuous scale.)

Note: Take 10 decimal places after the ".", if the answer is a fraction, enter the fraction (a/b). Use a period (.) not a comma (,) for decimals.

a) If a specimen is acceptable only if its hardness is between 67 and 71, what is the probability that a randomly chosen specimen has an acceptable hardness?

P(67≤X≤71) = ?

b) If the acceptable range of hardness is (69−c,69+c), for what value of c would 96% of all specimens have acceptable hardness?

c = ?

c) If the acceptable range is as in part (a) and the hardness of each of 6 randomly selected specimens is independently determined, what is the probability that at most 4 of 6 independently selected specimens have a hardness of this acceptable range.

Y= the number of specimens have a hardness of this acceptable range.

P(Y≤4) = ?

because that’s us why