a) 75% in the middle leaves 1/2 of 25% or 12.5% on each tail.
The middle 75% will go from 12.5% to 87.5%
The z-score associated with 87.5% or an 0.875 probability is 1.15
to find the XR associated with that probability use formula:
z = (xR - mean)/SD, which for this problem z = (x - 20)/10
so since z = 1.15 at 0.875:
1.15 = (xR - 20)/10
and xR = 31.5
because normal distribution is symmetric, z-score associated with 12.5% or 0.125 probability is -1.15.
so
-1.15 = (XL - 20)/10 = 8.5
8.5 < X < 31.5
b) 84% in the middle, leaves 1/2 of 16% or 8% on each tail.
The middle 84% will go from 8% to 92%
Proceed as in part a) finding the z-score associated with 92% or 0.92 probability, using the negative of that z-score for the 0.08 probability, then solving for XR and XL using the standardized z-score equation.
