Question

What is the probability that Upper X less than 95 or Upper X greater than 105​? The probability that Upper X less than 95 or Upper X greater than 105 is nothing. ​(Round to four decimal places as​ needed.)

Answer 1

Answer:

$P(X$

And for the other case if we use the z score and the complement rule we have:

$P(X>105) =P(Z>\frac{105-100}{10}) =1-P(Z$

And we can find the probability of interest like this:

$P(X105)=P(X105) -P(X105)$And replacing we got:

$P(X105)=0.309+0.309-0= 0.618$

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:

We assume that the distribution is $X \sim N(100,10)$  

Where $\mu=100$ and $\sigma=10$

We are interested on this probability

$P(X105)$

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 find the individual probabilities we got:

$P(X$

And for the other case if we use the z score and the complement rule we have:

$P(X>105) =P(Z>\frac{105-100}{10}) =1-P(Z$

And we can find the probability of interest like this:

$P(X105)=P(X105) -P(X105)$And replacing we got:

$P(X105)=0.309+0.309-0= 0.618$