# Type-I and Type-II errors example explanation

In the following example, can someone help understand how the value for $$k$$ (highlighted in yellow) is derived below? How is $$z_\alpha$$ introduced in there?

We want to find the value $$k$$ such that it is exceeded by the sample mean with probability $$\alpha$$ if the null is true.
In the display, the second line subtracts $$\mu_0$$ and divides by $$1/\sqrt{n}$$ to get a standard normal r.v. under the null. By definition, a r.v. exceeds its $$1-\alpha$$-quantile with probability $$\alpha$$. $$z_\alpha$$ is this quantile. Hence, $$\frac{k-\mu_0}{1/\sqrt{n}}=z_\alpha$$ Now, solve for $$k$$.
• Thanks for the explanation. Can you please give me a reference where I can read more about quantiles and the fact that random variable exceeds its $1-\alpha$-quantile with probability $\alpha$ and that we denote this quantile by $z_\alpha$? Much appreciated! – cbro Dec 17 '18 at 11:49
• en.wikipedia.org/wiki/Quantile has further discussion. $z_\alpha$ is just notation. – Christoph Hanck Dec 17 '18 at 12:48