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?

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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 $k$.

  • $\begingroup$ 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! $\endgroup$ – cbro Dec 17 '18 at 11:49
  • $\begingroup$ en.wikipedia.org/wiki/Quantile has further discussion. $z_\alpha$ is just notation. $\endgroup$ – Christoph Hanck Dec 17 '18 at 12:48

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