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 68 and standard deviation 3. (Rockwell hardness is measured on a continuous scale).(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-c, 70+c), for what value of c would 95% of all specimens have acceptable hardness?

Answer 1

Explanation:

Given that 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.

Let X be the hardness

X is N(68,3)

`a) P(67<x<75) =\\P((67-68)/(3) <Z<(75-68)/(3) \\=P(-0.33<z<2.33)`

This area lies on both sides of the mean. Hence we add the corresponding prob values form table

= 0.1293+0.4901=0.6194