2 Simple comparative experiments
The simplest experiment compares two treatments. It is worth treating carefully, because every idea of the later sections appears here in its plainest form: a model for the observations, an estimate of experimental error, a test and a confidence interval, a check of the assumptions, a choice of sample size, and the gain from blocking.
Example 2.0.1. A chemical engineer compares the yield (%) of a process run with two catalysts. Sixteen batches of raw material are available; eight are chosen at random for catalyst \(A\) and the other eight are run with catalyst \(B\), in random order. The yields are
| batch | catalyst \(A\) | catalyst \(B\) |
| \(1\) | \(91.5\) | \(89.2\) |
| \(2\) | \(94.2\) | \(91.0\) |
| \(3\) | \(92.2\) | \(90.5\) |
| \(4\) | \(95.4\) | \(93.2\) |
| \(5\) | \(91.8\) | \(88.9\) |
| \(6\) | \(89.1\) | \(87.6\) |
| \(7\) | \(94.7\) | \(90.1\) |
| \(8\) | \(89.2\) | \(89.9\) |
| \(\bar y_i\) | \(92.2625\) | \(90.0500\) |
| \(s_i^2\) | \(5.6741\) | \(2.7286\) |
Catalyst \(A\) gives the higher mean yield in these data. Is the difference real, or could it have arisen from batch-to-batch variation alone?
2.2 The two-sample \(t\) test
2.3 Unequal variances
2.4 Checking the assumptions
2.5 Choice of sample size
2.6 The paired comparison design
2.7 Practice problems
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