# Statistics for Environmental Engineers

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The standard error of the difference is:

^y,У2 — Vo1Bi + „2

which is estimated with v nl + n2 2 degrees of freedom. Assuming the variances before and after the intervention are the same, s — s2 58 and therefore spool — 58.

For a — 0.05, n1 — 10, and assuming v — 10 + n2 — 2 > 30, t0.05 1.70. The sample size n2 must be large enough that:

1.7(58) 1 + 1< 25

10 n2

This condition is impossible to satisfy. Even with n2 — ~, the left-hand side of the expression gets only as small as 31.2.

The managers should have asked the sampling design question before the pre-change survey was made, and when a larger pre-change sample could be taken. A sample of n1 — n2 ~ 32 would be about right.

### What about Type II Error?

So far we have mentioned only the error that is controlled by selecting a. That is the so-called type I error, which is the error of declaring an effect is real when it is in fact zero. Setting a — 0.05 controls this kind of error to a probability of 5%, when all the assumptions of the test are satisfied.

Protecting only against type I error is not totally adequate, however, because a type I error probably never occurs in practice. Two treatments are never likely to be truly equal; inevitably they will differ in some respect. No matter how small the difference is, provided it is non-zero, samples of a sufficiently large size can virtually guarantee statistical significance. Assuming we want to detect only differences that are of practical importance, we should impose an additional safeguard against a type I error by not using sample sizes larger than are needed to guard against the second kind of error.

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