Chris M. answered • 05/02/20
Former PhD. student (8.5+ years) M.B.A.
In short:
You change your values to Z scores (This is the z score formula: z = (x-μ)/σ) and then you pay attention to whether you are being asked for the probability x lies above or below a number or the probability that it lies between two numbers.
You can use your graphing calculator or websites to get helpful pictures of what this looks like.
Here is a longer answer
When dealing with probabilities, the maximum score is 1. You have a total area underneath the line (usually called a 'curve' that is equal to the maximum...the area will be one.
You are only trying to find part of this area, so your answers will be less than one.
Since this is a "normal" distribution, you can find the areas using a probability distribution. You don't need to use anything as complicated as calculus. People have already figured out the areas under all the parts of this curve already. You don't need to do that.
The first thing you will do is to change your numbers to "z scores". Z scores are used to help you find the answer as if your distribution were a standard distribution instead of one with a unique mean and standard deviation.
z scores are just the x value you are given minus the mean (x-μ) over the standard deviation. (σ)
(The symbol--> μ is called mu and it represents the arithmetic mean) (This symbol --> σ is known as sigma)
In the most simple case, you are given 865. You are asked to find the probability that x will be less than 865. This is the same as asking the area to the left of 865. If you use the z score formula, you find
z=-1.35=(865-1000)/100 You need to find the area/probability under -1.35z
You can use your calculator, a table, excel, or a page such as https://www.zscorecalculator.com/ to get the probability from this point. Your probability will be 0.0885 and your calculator or a website should give you a nice image of this to help you understand it better.
With A, the question is a litle more complicated. You can find the area under 1155 as before. If you do this, you know that .9394292 of the area lies under 1155. But the question doesn't just ask you what is the probability x will be less than 1155. It asks what the probability is that x is less than 1155 and greater than 1000. However, you can find that there is a .5 probability that x is less than 1000 and subtract .5 from .9394292 to find that there is a .0.4394292 probability that x lies betweeen 1000 and 1155
The terms left tail(ed), right tail(ed) and two-tailed may be confusing, so I did not use them, but you will need to make sure you understand them for your statistics courses.
