Question 1188411
mean = 200
standard deviation = 25


z-score = (x - m) / s


z is the z-score
x is he raw score
m is the mean
s is the standard deviation


when x = 150, z = (150 - 200) / 25 = -2
when x = 275, z = (275 - 200 / 25 = 3


use a calculator or look at the normal distribution z-score table to get the area to the left of those z-scores.


the table i used is at <a href = "https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/Standard%20Normal%20Distribution%20Table.pdf" target = "_blank">https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/Standard%20Normal%20Distribution%20Table.pdf</a>


this table gives area to the left of the z-score.


area to the left of z-score of -2 = .02275
area to the left of z-score  of 3 = .99865


subtract the smaller area from the larger area to get area in between = .97590.


that's the ratio of the area under the normal distribution curve that is between those z-scores.


multiply that by 100 to get 97.590%.


you can do the same thing using a calculator.


using my ti-85 plus, i get:


area to the left of z-score of -2 = .022750062
area to the left of z-score of 3 = .9986500328


subtract the smaller area from the larger area to get area in between = .9758999708.


using the ti-85 plus calculator, i can get the area in between without having to find the area to the left of each z-score.


the area in between -2 and 3 is equal to .9758999708.


this was done in one shot, rather than 2.


there are online calculator that do the same for you.


one such calculator can be found at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blank">https://davidmlane.com/hyperstat/z_table.html</a>


here's he result from using that calculator.


<img src = "http://theo.x10hosting.com/2021/120401.jpg" >


that calculator could also have found the area in between from the raw scores.


in that case, you need to enter the mean and standard deviation, rathern than 0 for the mean and 1 for the standard deviation, as you needed to do with z-scores.


the results of entering the mean and standard deviation and raw scores is shown below.


<img src = "http://theo.x10hosting.com/2021/120402.jpg" >