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 https://www.rit.edu/academicsuccesscenter/sites/rit.edu.academicsuccesscenter/files/documents/math-handouts/Standard%20Normal%20Distribution%20Table.pdf
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 https://davidmlane.com/hyperstat/z_table.html
here's he result from using that calculator.
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.
