Q.For testing $H_0 : \mu = \mu_0$ against $H_1 : \mu < \mu_0$, what is the critical value at $\alpha = 0.01$ ?

A left-tailed test at $\alpha = 0.01$ puts all $1\%$ in the lower tail. The $z$ value cutting off the lower $1\%$ is $-2.33$.

Step 1 — Identify the tail. $H_1 : \mu < \mu_0$ ⟹ left- (lower-) tailed test; the rejection region is in the left tail only.

Step 2 — Place the significance level. All of $\alpha = 0.01$ lies in the left tail, so we need $z$ with $P(Z < z) = 0.01$.

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