Confidence Intervals and Hypothesis Tests for the Difference between Two Population Means

Constructing a CI

• Parameter: \(\mu_1 - \mu_2\)
  • Independent, random, normal samples
• Estimate: \(\bar{x_1} - \bar{x_2}\)
• Standard error: \(\text{s.e.} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\)
• Pooled standard error: \(\text{s.e.} = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\)
• Pooled sample standard devation:

$$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$

If sample std devs similar, pool. Otherwise, if larger std dev from group with larger sample, pooled acceptable, conservative. If the smaller std dev came from larger same produces a misleading smaller p-value.