## One sample t test

Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ and unknown variance $\sigma^2$.

The hypothesis 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

`$t = \dfrac{\overline{x}-\mu_0}{SE(\overline{x})}$`

where,

`$\overline{x}=\dfrac{1}{n} \sum x_i$`

is the sample mean,`$SE(\overline{x})=\dfrac{s}{\sqrt{n}}$`

is the standard error of mean,`$s^2 =\dfrac{1}{n-1} \sum (x_i -\overline{x})^2$`

is the sample variance.

The test statistic $t$ follows $t$ distribution with $n-1$ degrees of freedom.