Two sample t test with unknown and equal variances
Let $\overline{x}_1$ be the sample mean and $s_1$ be the sample standard deviation of a random sample of size $n_1$ from a population with mean $\mu_1$ and variance $\sigma^2_1$.
Let $\overline{x}_2$ be the sample mean and $s_2$ be the sample standard deviation of a random sample of size $n_2$ from a population with mean $\mu_2$ and variance $\sigma^2_2$.
Suppose the variances $\sigma^2_1$ and $\sigma^2_2$ are unknown but equal.
The hypothesis testing problem can be set up as:
| Situation | Hypothesis Testing Problem |
|---|---|
| Situation A : | $H_0: \mu_1=\mu_2$ against $H_a : \mu_1 < \mu_2$ (Left-tailed) |
| Situation B : | $H_0: \mu_1=\mu_2$ against $H_a : \mu_1 > \mu_2$ (Right-tailed) |
| Situation C : | $H_0: \mu_1=\mu_2$ against $H_a : \mu_1 \neq \mu_2$ (Two-tailed) |
Formula
The test statistic for testing above hypothesis is
$t=\frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{SE(\overline{x}_1-\overline{x}_2)}= \frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$
where
-
$SE(\overline{x}_1-\overline{x}_2) = s_p\sqrt{\frac{1}{n_1}+ \frac{1}{n_2}}$is the standard error of difference between means, -
$s_p=\sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}$is the pooled standard deviation.
The test statistic $t$ follows Students’ $t$ distribution with $n_1+n_2-2$ degrees of freedom.
