Calculation of Standard Deviation of Sample Mean

This is another statistics problem that I have, which I cannot make sense of:

An IQ-test is normal to $\mu = 100$ and $\sigma = 10$. What is the standard deviation of the sample mean of a sample with size $n = 50$?

We know the mean is $\mu = n \times p$ which we solve for $n=\frac{100}{p}$ and substitute in $Var(X) = n \times p(1-p)$ which is $\sigma^2$ but yet I don't seem to be able to obtain the value of the standard deviation $\sigma$ using this way. The supposed solution of this problem is $1.4142$.

• I did not come up with this by myself, my textbook actually does apply this formula $\sqrt{n} \times \frac{\overline{X}-\mu}{\sigma}$ to this problem. One can obtain $\mu$ with $n \times p$ if it is not given, but here it is given so that is not relevant. And the standard deviation $\sigma$ can be gotten by $\sqrt{np \times (1-p)}$ which is what I am trying to follow here to find it. – Khaled Apr 6 '18 at 2:48