Question

Assume the random variable X is normally distributed with mean mu equals 50 and standard deviation sigma equals 7. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded. Upper P (34 less than Upper X less than 58 )Which of the following normal curves corresponds to Upper P (34 less than Upper X less than 58 )​?

Answer 1

`P(34<X<58)=P((34-\mu)/(\sigma)<(X-\mu)/(\sigma)<(58-\mu)/(\sigma))=P((34-50)/(7)<Z<(58-50)/(7))=P(-2.286<z<1.143)`

And we can find this probability with this difference:

`P(-2.286<z<1.143)=P(z<1.143)-P(z<-2.286)`

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

`P(-2.286<z<1.143)=P(z<1.143)-P(z<-2.286)=0.873-0.0111=0.862`

The figure attached illustrate the problem 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 variable of interest of a population, and for this case we know the distribution for X is given by:

`X \sim N(50,7)`

Where
`\mu=50` and
`\sigma=7`

We are interested on this probability

`P(34<X<58)`

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(34<X<58)=P((34-\mu)/(\sigma)<(X-\mu)/(\sigma)<(58-\mu)/(\sigma))=P((34-50)/(7)<Z<(58-50)/(7))=P(-2.286<z<1.143)`

And we can find this probability with this difference:

`P(-2.286<z<1.143)=P(z<1.143)-P(z<-2.286)`

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

`P(-2.286<z<1.143)=P(z<1.143)-P(z<-2.286)=0.873-0.0111=0.862`

The figure attached illustrate the problem for this case.