# Statistics for Environmental Engineers

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n = 2q2(Zq/2 + Zp) = 2(0.06)21.96 + (1.280)2 = 1 21 ^ 2 A2    0.252    .

Setting the probability of the type I and type II errors may be difficult. Typically, a is specified first. If declaring the two treatments to differ significantly will lead to a decision to conduct further expensive research or to initiate a new and expensive form of treatment, then a type I error is serious and it should be kept small (a = 0.01 or 0.02). On the other hand, if additional confirmatory testing is to be done in any case, as in routine monitoring of an effluent, the type I error is less serious and a can be larger.

Also, if the experiment was intended primarily to add to the body of published literature, it should be acceptable to increase a to 0.05 or 0.10.

Having specified a, the investigator needs to specify в, or 1 — fi. Cohen (1969) suggests that in the context of medical treatments, a type I error is roughly four times as serious as a type II error. This implies that one should use approximately в = 4a so that the power of the test is 1 — в = 1 — 4a. Thus, when a = 0.05, set 1 — в = 0.80, or perhaps less.

### Sample Size for Assessing the Equivalence of Two Means

The previous sections dealt with selecting a sample size that is large enough to detect a difference between two processes. In some cases we wish to establish that two processes are not different, or at least are close enough to be considered equivalent. Showing a difference and showing equivalence are not the same problem.

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