# Thread: Probabilities and Normal Distribution Question

1. ## Probabilities and Normal Distribution Question

I can't seem to find the answer for this question; so far I've tried weighted averages and finding the closest equivalent z-score for the percentages, but I'm at a loss.

The heights of individuals in a mall are normally distributed. If 66% of the shoppers are shorter than 160 cm and 82% of the shoppers are shorter than 170 cm, determine the mean height of the shoppers. Round your answer to the nearest hundredth of a centimeter if necessary.

2. ## Re: Probabilities and Normal Distribution Question

There are two parameters to solve for and you've got two conditions.

Let $\Phi(x)$ be the CDF of the standard normal distribution

$\Phi\left(\dfrac{160-\mu}{\sigma}\right) = 0.66$

$\Phi\left(\dfrac{170-\mu}{\sigma}\right) = 0.82$

$\dfrac{160-\mu}{\sigma} = \Phi^{-1}(0.66) = 0.412463$

$\dfrac{170-\mu}{\sigma} = \Phi^{-1}(0.82) = 0.915365$

It's a simple algebra problem now which I leave for you to solve.