For $n=20$ samples of normal distribution with variance $10$, what should be $\alpha$, so the length of the confidence interval will be exactly $2$?
So the calculation is:
$$2=2z_{1-\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}=2z_{1-\frac{\alpha}{2}}\frac{\sqrt{10}}{\sqrt{20}}=2z_{1-\frac{\alpha}{2}}\sqrt{\frac{1}{2}}\\z_{1-\frac{\alpha}{2}}=\sqrt{2} $$
BUT! There's a problem - $\sqrt 2 > 1$
Why is it happening and how to fix it?
Your $z$-value can be greater than $1$ - you're confusing them with probabilities. You need to look in the $z$-table and find a corresponding choice of $\alpha$ for $\sqrt{2} \approx 1.41$ and set that to $1-\frac{\alpha}{2}$.