# Statistics for Environmental Engineers

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We can specify the equivalence interval such that в = ви = -вь. When the common variance a2 is known, the rule is to reject H0 in favor of H1 if:

-в + zaOyi-y2 — y1 — y2 — в — zaOyi-y2

The approximate sample size for the case where n1= n2= n is:

2a2(za + zв)2, , n = f — + 1

(в-A)2

в defines (a priori) the practical equivalence limits, or how close the true treatment means are required to be before they are declared equivalent. A is the true difference between the two treatment means under which the comparison is made.

Stein and Dogansky (1999) give an iterative solution for the case where a different sample size will be taken for each treatment. This is desirable when data from the standard process is already available.

In the interval hypothesis, the type I error rate (a) denotes the probability of falsely declaring equivalence. It is often set to a = 0.05. The power of the hypothesis test (1 ) is the probability of correctly declaring equivalence. Note that the type I and type II errors have the reverse interpretation from the classical hypothesis formulation.

Example 23.5

A standard process is to be compared with a new process. The comparison will be based on taking a sample of size n from each process. We will consider the two process means equivalent if they differ by no more than 3 units (в = 3.0), and we wish to determine this with risk levels a = 0.05, в = 0.10, a = 1.8, when the true difference is at most 1 unit (A = 1.0). The sample size from each process is to be equal. For these conditions, z0.05= 1.645 and z0.10= 1.28, and:

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