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.
