 7. Product and Process Comparisons
7.3. Comparisons based on data from two processes
7.3.1. Do two processes have the same mean?

## Confidence intervals for differences between means

Definition of confidence interval for difference between population means Given two random samples, $$Y_1, \, \ldots, \, Y_N \,\,\,\,\, \mbox{ and } \,\,\,\,\, Z_1, \, \ldots, \, Z_N$$

from two populations, two-sided confidence intervals with $$100(1-\alpha)$$ % coverage for the difference between the unknown population means, $$\mu_1$$ and $$\mu_2$$, are shown in the table below. Relevant statistics for paired observations and for unpaired observations are shown elsewhere.

Two-sided confidence intervals with $$100(1-\alpha)$$ % coverage for $$\mu_1 - \mu_2$$:

 $$\mu_1 - \mu_2 \,\,\, (\mbox{where } \sigma_1 = \sigma_2)$$ $$\bar{d} \pm t_{1-\alpha/2, \, N-1} \frac{s_{d}}{\sqrt{N}}$$

 $$\mu_1 - \mu_2 \,\,\, (\mbox{where } \sigma_1 = \sigma_2)$$ $$\bar{Y} - \bar{Z} \pm t_{1-\alpha/2, \, N_1 + N_2 - 2} \,\, s \sqrt{\frac{1}{N_1} + \frac{1}{N_2}}$$ $$\mu_1 - \mu_2 \,\,\, (\mbox{where } \sigma_1 \ne \sigma_2)$$ $$\bar{Y} - \bar{Z} \pm t_{1-\alpha/2, \, N_1 + N_2 - 2} \, \sqrt{\frac{s_{1}^{2}}{N_1} + \frac{s_{2}^{2}}{N_2}}$$

Interpretation of confidence interval One interpretation of the confidence interval for means is that if zero is contained within the confidence interval, the two population means are equivalent. 