Question

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

Answer 1

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%