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

## Analysis of paired observations

Definition of paired comparisons Given two random samples, $$Y_1, \, \ldots, \, Y_N \,\,\,\,\, \mbox{ and } \,\,\,\,\, Z_1, \, \ldots, \, Z_N$$

from two populations, the data are said to be paired if the $$i$$-th measurement on the first sample is naturally paired with the $$i$$-th measurement on the second sample. For example, if $$N$$ supposedly identical products are chosen from a production line, and each one, in turn, is tested with first one measuring device and then with a second measuring device, it is possible to decide whether the measuring devices are compatible; i.e., whether there is a difference between the two measurement systems. Similarly, if "before" and "after" measurements are made with the same device on $$N$$ objects, it is possible to decide if there is a difference between "before" and "after"; for example, whether a cleaning process changes an important characteristic of an object. Each "before" measurement is paired with the corresponding "after" measurement, and the differences, $$d_i = Y_i - Z_i \,\,\,\,\, (i = 1, \, \ldots, \, N) \, ,$$ are calculated.

Basic statistics for the test The mean and standard deviation for the differences are calculated as $$\bar{d} = \frac{1}{N} \sum_{i=1}^N d_i$$ and $$s_d = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (d_i - \bar{d})^2}$$ with $$\nu = N-1$$ degrees of freedom.
Test statistic based on the $$t$$ distribution The paired-sample $$t$$ test is used to test for the difference of two means before and after a treatment. The test statistic is: $$t = \frac{\bar{d}}{s_d / \sqrt{N}} \, .$$ The hypotheses described on the foregoing page are rejected if:
1. $$|t| \ge t_{1-\alpha/2, \, \nu}$$
2. $$t \ge t_{1-\alpha, \, \nu}$$
3. $$t \le t_{\alpha, \, \nu}$$
where for hypothesis (1), $$t_{1-\alpha/2, \, \nu}$$ is the $$1-\alpha/2$$ critical value from the $$t$$ distribution with $$\nu$$ degrees of freedom and similarly for cases (2) and (3). Critical values can be found in the t table in Chapter 1. 