You can encounter this type of questions during the interview process for Data Scientist positions. So the question can be like that:

**Question**: Assume that a process follows a normal distribution with **mean** **50** and that we have observed that the **probability** **to exceed the value 60 is 5%**. What is the standard deviation of the distribution?

**Solution**:

\(P(X \geq 60) = 0.05\)

\(1- P(X < 60) = 0.05\)

\(P(X < 60) = 0.95\)

\(P(\frac{X-50}{\sigma} < \frac{60-50}{\sigma}) = 0.95\)

\(P(\frac{X-50}{\sigma} < \frac{10}{\sigma}) = 0.95\)

\(Z(\frac{10}{\sigma})= 0.95\)

But form the Standard Normal Distribution we know that the \(Z(1.644854)=0.95\) (`qnorm(0.95) = 1.644854`

), Thus,

\(\frac{10}{\sigma} = 1.644854\)

\(\sigma = 6.079567\)

Hence the **Standard Deviation is 6.079567**. We can confirm it by running a simulation in R estimating the probability of the Normal(50, 6.079567) to exceed the value 60:

set.seed(5) sims<-rnorm(10000000, 50, 6.079567 ) sum(sims>=60)/length(sims)

And we get:

[1] 0.0500667

As expected, the estimated probability for our process to exceed the value 60 is 5%.