Two Population Inference

Is Method A better than Method B? Do men earn more than women? We compare two groups to find out.

Comparing Two Means

We are testing the difference between two population means ($\mu_1 - \mu_2$). Usually, our Null Hypothesis is that there is no difference.

$H_0: \mu_1 - \mu_2 = 0$ (or $\mu_1 = \mu_2$)

We use the Z-test. This is rare in practice but good for understanding.

$$ Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} $$

If we assume the variances are equal ($\sigma_1^2 = \sigma_2^2$), we pool the sample variances.

$$ t = \frac{(\bar{x}_1 - \bar{x}_2)}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} $$
Where $s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}$

Degrees of freedom: $df = n_1 + n_2 - 2$.

We use the unpooled t-test (Welch's t-test). The formula for degrees of freedom is complex, but software handles it.

$$ t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} $$

Q1: When comparing two means with unknown but equal variances, we calculate a ______ variance.