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

7.3.1.2.

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 \(\mu_1 - \mu_2\):

Paired observations

$$ \mu_1 - \mu_2 \,\,\, (\mbox{where } \sigma_1 = \sigma_2) $$

$$ \bar{d} \pm t_{1-\alpha/2, \, N-1} \frac{s_{d}}{\sqrt{N}} $$

Unpaired observations

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

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.