One Sample Z Test For Population Mean
Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ and known variance $\sigma^2$. Let $\overline{x}=\frac{1}{n} \sum X_i$ be the sample mean.
The hypothesis testing problem can be setup as :
| Situation | Hypothesis Testing Problem |
|---|---|
| Situation A: | $H_0: \mu=\mu_0$ against $H_a : \mu < \mu_0$ (Left-tailed) |
| Situation B: | $H_0: \mu=\mu_0$ against $H_a : \mu > \mu_0$ (Right-tailed) |
| Situation C: | $H_0: \mu=\mu_0$ against $H_a : \mu \neq \mu_0$ (Two-tailed) |
Formula
The test statistic under $H_0:\mu=\mu_0$ is
$Z = \frac{\overline{x}-\mu_0}{SE(\overline{x})}$
The test statistic $Z$ follows standard normal distribution $N(0,1)$.
where,
$\overline{x} =\dfrac{1}{n}\sum x_i$is the sample mean,$SE(\overline{x})=\dfrac{\sigma}{\sqrt{n}}$$\sigma$is the population standard deviation.
