Question

The length of life of an instrument produced by a machine has a normal ditribution with a mean of 12 months and standard deviation of 2 months. Find the probability that an instrument produced by this machine will last

Answer 1

a)
`P(X<7)=P((X-\mu)/(\sigma)<(7-\mu)/(\sigma))=P(Z<(7-12)/(2))=P(z<-2.5)`

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

`P(z<-2.5)=0.0062`

b)
`P(7<X<12)=P((7-\mu)/(\sigma)<(X-\mu)/(\sigma)<(12-\mu)/(\sigma))=P((7-12)/(2)<Z<(12-12)/(2))=P(-2.5<z<0)`

And we can find this probability with this difference:

`P(-2.5<z<0)=P(z<0)-P(z<-2.5)`

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

`P(-2.5<z<0)=P(z<0)-P(z<-2.5)=0.5-0.0062=0.494`

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".

a) less than 7 months.

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

`X \sim N(12,2)`

Where
`\mu=12` and
`\sigma=2`

We are interested on this probability

`P(X<7)`

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<7)=P((X-\mu)/(\sigma)<(7-\mu)/(\sigma))=P(Z<(7-12)/(2))=P(z<-2.5)`

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

`P(z<-2.5)=0.0062`

b) between 7 and 12 months.

`P(7<X<12)=P((7-\mu)/(\sigma)<(X-\mu)/(\sigma)<(12-\mu)/(\sigma))=P((7-12)/(2)<Z<(12-12)/(2))=P(-2.5<z<0)`

And we can find this probability with this difference:

`P(-2.5<z<0)=P(z<0)-P(z<-2.5)`

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

`P(-2.5<z<0)=P(z<0)-P(z<-2.5)=0.5-0.0062=0.494`