## 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.