Number of passengers who arrive at the platform in an Amtrack train station for the 2 pm train on a Saturday is a random variable with a mean of 180 and a standard deviation of 25. With what probability can we assert that there will be between 80 and 280 passengers?

And we can find this probability with this difference:

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

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 number of passengers who arrive at the platform of a population, and for this case we know the following info

Where and

We are interested on this probability

For this case we can assume that the random variable is normally distributed.

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

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

And we can find this probability with this difference:

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