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

Answer:

$P(35<X<65)=P(\frac{35-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{65-\mu}{\sigma})=P(\frac{35-50}{7}<Z<\frac{65-35}{7})=P(-2.143<z<4.286)$

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

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

Step-by-step 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:

$z=\frac{x-\mu}{\sigma}$

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

$P(35<X<65)=P(\frac{35-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{65-\mu}{\sigma})=P(\frac{35-50}{7}<Z<\frac{65-35}{7})=P(-2.143<z<4.286)$

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

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