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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.
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