# The average height for males is normally distributed with a mean of 70in. and a standard deviation of 4.5in. If a randomly selected male is taller than 25% of males, about how tall is he?

tiburtius | High School Teacher | (Level 2) Educator

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Probability density function for normal random variable is

`f(x)=1/(sigma sqrt(2pi))e^(-(x-mu)^2/(2sigma^2))`

`f(x)=1/(4.5sqrt(2pi))e^(-(x-70)^2/(2cdot4.5^2))`

You are looking for 25th percentile. For that you need to find solution to the equation

`Phi(x)=0.25`                                                                  (1)

where `Phi` is cumulative distribution function of your random variable that is

`Phi(x)=int_-oo^xf(x)dx`

Hence, you need to find solution to the equation

`1/(4.5sqrt(2pi))int_-oo^x e^(-(x-70)^2/(2cdot4.5^2))=0.25`                                          (2)

Since there are no elementary function corresponding to `Phi` the equation must be solved numerically.

Solution to equation (2) is `x approx 66.9648`.

So the man is about 70 inches (66.9648in to be exact) tall.