Question

A sales completion team, aiming to reduce the shipment time of urgent orders, studies the current process and finds that the current shipment time has a mean of 4 days with a standard deviation of 1.5 days. Industry expectations are for urgent shipments to be delivered between 1 and 5 days.

Answer 1

`P(1<X<5)=P((1-\mu)/(\sigma)<(X-\mu)/(\sigma)<(5-\mu)/(\sigma))=P((1-4)/(1.5)<Z<(5-4)/(1.5))=P(-2<z<0.667)`

And we can find this probability with thi difference:

`P(-2<z<0.667)=P(z<0.667)-P(z<-2)`

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

`P(-2<z<0.667)=P(z<0.667)-P(z<-2)=0.748-0.023=0.725`

So then we expect about 72.5% of the current shipment times between 1 and 5 days

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 current shipment time of a population, and for this case we know the distribution for X is given by:

`X \sim N(4,1.5)`

Where
`\mu=4` and
`\sigma=1.5`

We are interested on this probability

`P(1<X<5)`

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(1<X<5)=P((1-\mu)/(\sigma)<(X-\mu)/(\sigma)<(5-\mu)/(\sigma))=P((1-4)/(1.5)<Z<(5-4)/(1.5))=P(-2<z<0.667)`

And we can find this probability with thi difference:

`P(-2<z<0.667)=P(z<0.667)-P(z<-2)`

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

`P(-2<z<0.667)=P(z<0.667)-P(z<-2)=0.748-0.023=0.725`

So then we expect about 72.5% of the current shipment times between 1 and 5 days