Answer:
 P [  1689  ≤   X  ≤  2267 ]  = 54,88 %
Step-by-step explanation:
Normal Distribution
Mean        μ₀  =  1730
Standard Deviation      σ  = 257
We need to calculate  z scores for the values   1689     and      2267
We apply formula for z scores
z =  ( X -  μ₀ ) /σ
X = 1689     then
z = (1689 - 1730)/ 257      ⇒ z = - 41 / 257
z  = -  0.1595 
And from z table we get  for  z =  - 0,1595
We have to interpolate
         - 0,15          0,4364
         - 0,16          0,4325
Δ  =   0.01           0.0039
 0,1595  -  0,15  =  0.0095
By rule of three
0,01                  0,0039
0,0095                 x ??      x  =  0.0037
And    0,4364  -  0.0037  = 0,4327
Then    P [ X ≤ 1689 ]  =  0.4327     or    P [ X ≤ 1689 ]  = 43,27 %
And for the upper limit  2267  z  score will be
z  =  ( X - 1730 ) / 257       ⇒  z =  537 / 257
z  =  2.0894 
Now from z table   we find  for score   2.0894
We interpolate and assume  0.9815
P [ X ≤ 2267 ]  =  0,9815
Ths vale already contains th value of   P [ X ≤ 1689 ]  =  0.4327
Then we subtract  to get    0,9815  -  0,4327   = 0,5488
Finally
P [ 1689  ≤   X  ≤  2267 ]  =  0,5488  or  P [  1689  ≤   X  ≤  2267 ]  = 54,88 %