Question

Heated-treated parts are specified to have a hardness of at least 45 (lower specification limit). Any parts that are below 45 are too soft and have to be heat treated again at great expense. The hardness of heat-treated parts actually produced follows a normal distribution with a mean of 48.5 with a standard deviation of 2. (Note: sketch of the distribution helps the set up your calculation). What is the probability that a part is below the minimum specified hardness

Answer 1

`P(X<45)=P((X-\mu)/(\sigma)<(45-\mu)/(\sigma))=P(Z<(45-48.5)/(2))=P(z<-1.75)`

And we can find this probability using the normal standard table or excel and we got:

`P(z<-1.75)=0.04`

The figure shows the calculation for this case.

Explanation:

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:

`X \sim N(48.5,2)`

Where
`\mu=48.5` and
`\sigma=2`

We are interested on this probability

`P(X<45)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X<45)=P((X-\mu)/(\sigma)<(45-\mu)/(\sigma))=P(Z<(45-48.5)/(2))=P(z<-1.75)`

And we can find this probability using the normal standard table or excel and we got:

`P(z<-1.75)=0.04`

The figure shows the calculation for this case.