Question

Assume the random variable X is normally distributed with meanmu equals 50μ=50and standard deviationsigma equals 7σ=7.Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.Upper P left parenthesis 35 less than Upper X less than 65 right parenthesisP(35

Answer 1

`P(35<X<65)=P((35-\mu)/(\sigma)<(X-\mu)/(\sigma)<(65-\mu)/(\sigma))=P((35-50)/(7)<Z<(65-35)/(7))=P(-2.143<z<4.286)`

And we can find this probability with the following difference and using the normal standard table or excel we have:

`P(-2.143<z<4.286)=P(z<4.286)-P(z<-2.143)=0.9999-0.0161=0.9839`

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(35<X<65)`

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(35<X<65)=P((35-\mu)/(\sigma)<(X-\mu)/(\sigma)<(65-\mu)/(\sigma))=P((35-50)/(7)<Z<(65-35)/(7))=P(-2.143<z<4.286)`

And we can find this probability with the following difference and using the normal standard table or excel we have:

`P(-2.143<z<4.286)=P(z<4.286)-P(z<-2.143)=0.9999-0.0161=0.9839`