Mean and standard deviation from percentage values - Normal distribution

Question

Suppose in a sample of 500 observations from a normal distribution with a mean μ and a standard deviation σ, you are told that 35% of the observations are less than 2.1 and 55% of the observations are less than 3.6. Estimate μ and σ.

Answer 1

This just needs a little bit of algebra. The Z value for the first interval (-Inf 2.1) is -0.385 The Z value for the second interval (-Inf 3.6) is .126

The formula for the Z statistic is Z = (x - u)/σ

We have two equations -0.385σ = 2.1 - u and 0.126σ = 3.6 - u. There are just two unknowns. We can solve for x number of unknowns if we have x equaitons of those unknowns.

using algebra we solve for the two unknowns (u and σ)

Mu (mean or u) = 3.23 sd (sigma or σ) = 2.935