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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?
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| 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. |
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| 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:
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