If y is normal with μ = 200 and σ = 12, what is the probability that y will be between 186 and 210?

Normalizing the bounds,

y=186, z = (186-200) / 12 = - 1.17

y = 210, z = (210 - 200) / 12 = 0.83

We want the area of the standard normal between those. Looking it up in the cumulative table,

Ф (1.17) = 0.87900

Ф (0.83) = 0.79673

We need Ф (-1.17) which is 1 - Ф (1.17)

Then the probability we seek is

p = Ф (0.83) - Ф (-1.17) = Ф (0.83) + Ф (1.17) - 1 = 0.87900 + 0.79673 - 1 = 0.67573

Answer: 67.6%