Z-Test for μ (Known σ)¶
Overview¶
The one-sample Z-test tests hypotheses about the population mean when \(\sigma\) is known.
Hypotheses¶
\[
H_0: \mu = \mu_0 \quad \text{vs} \quad H_1: \mu \neq \mu_0 \text{ (or } \mu > \mu_0 \text{ or } \mu < \mu_0\text{)}
\]
Test Statistic¶
\[
Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}} \sim N(0,1) \text{ under } H_0
\]
Decision Rule (two-sided)¶
Reject \(H_0\) if \(|Z| > z_{\alpha/2}\).