Two sample Z test for means
Let $\overline{x}_1$ be the mean of a random sample of size $n_1$ from a population with mean $\mu_1$ and variance $\sigma^2_1$ and $\overline{x}_2$ be the mean of a random sample of size $n_2$ from a population with mean $\mu_2$ and variance $\sigma^2_2$.
The hypothesis testing problem can be setup as one of the three situations:
| 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 under $H_0 : \mu_1 = \mu_2$ is
$Z = \frac{(\overline{x}_1-\overline{x}_2)-(\mu_1-\mu_2)}{SE(\overline{x}_1-\overline{x}_2)}$
where,
-
$\overline{x}_1$is the sample mean of first sample of size $n_1$, -
$\overline{x}_2$is the sample mean of second sample of size $n_2$, -
$SE(\overline{x}_1-\overline{x}_2) = \sqrt{\frac{\sigma^2_1}{n_1}+ \frac{\sigma^2_2}{n_2}}$`,
The test statistic $Z$ follows standard normal distribution $N(0,1)$.
