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 )?

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:

[tex]X \sim N(50,7)[/tex]

Where [tex]\mu=50[/tex] and [tex]\sigma=7[/tex]

We are interested on this probability

[tex]P(34<X<58)[/tex]

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

[tex]z=\frac{x-\mu}{\sigma}[/tex]

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

[tex]P(34<X<58)=P(\frac{34-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{58-\mu}{\sigma})=P(\frac{34-50}{7}<Z<\frac{58-50}{7})=P(-2.286<z<1.143)[/tex]

And we can find this probability with this difference:

[tex]P(-2.286<z<1.143)=P(z<1.143)-P(z<-2.286)[/tex]

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

[tex]P(-2.286<z<1.143)=P(z<1.143)-P(z<-2.286)=0.873-0.0111=0.862[/tex]

The figure attached illustrate the problem for this case.