If a specimen is acceptable only if its hardness is between 63 and 75, what is the probability that a randomly chosen specimen has an acceptable hardness? (Round your answer to four decimal places.)

And we can find this probability with this difference:

And in order to find these probabilities we can using tables for the normal standard distribution, excel or a calculator.  

Step-by-step explanation:

Asuming this previous info: 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.

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the hardness of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability with this difference:

And in order to find these probabilities we can using tables for the normal standard distribution, excel or a calculator.  

17.08

correct me if i am wrong.