I need to understand how you arrive at your answer I can est

I need to understand how you arrive at your answer. I can estimate it easily, but need to be exact here.

On a normal distribution with a mean of 200 and a standard deviation of 50, what percentage of cases will fall between raw scores of 185 and 195?

Solution

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    185      
x2 = upper bound =    195      
u = mean =    200      
          
s = standard deviation =    50      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.3      
z2 = upper z score = (x2 - u) / s =    -0.1      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.382088578      
P(z < z2) =    0.460172163      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.078083585 or 7.808% [ANSWER]