# Statistics for Environmental Engineers

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The type II error is failing to declare an effect is significant when the effect is real. Such a failure is not necessarily bad when the treatments differ only trivially. It becomes serious only when the difference is important. Type II error is not made small by making a small. The first step in controlling type II error is specifying just what difference is important to detect. The second step is specifying the probability of actually detecting it. This probability (1 — в) is called the power of the test. The quantity в is the probability of failing to detect the specified difference to be statistically significant.

Figure 23.1 shows the situation. The normal distribution on the left represents the two-sided condition when the true difference between population means is zero (S 0). We may, nevertheless, with a probability of a/2, observe a difference d that is quite far above zero. This is the type I error. The normal distribution on the right represents the condition where the true difference is larger than d. We may, with probability в, collect a random sample that gives a difference much lower than d and wrongly conclude that the true difference is zero. This is the type II error.

The experimental design problem is to find the sample size necessary to assure that (1) any smaller sample will reduce the chance below 1 — в of detecting the specified difference and (2) any larger sample may increase the chance well above a of declaring a trivially small difference to be significant (Fleiss, 1981). The required sample size for detecting a difference in the mean of two treatments is:

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