## Interval of Acceptance

In testing for the outcome of the hypothesis, especially by simulation, we will "run" the hypothesis many times. The first few times may not be representative of the final outcome since it takes many runs to converge to the final and ultimate outcome. For a number of reasons, we may not have the luxury of waiting for or estimating the convergence. We may have to establish an interval around the likely outcome, called the interval of acceptance, within which we say that if an outcome falls anywhere in the interval of acceptance, then that outcome is "good enough." Now, if the objective is to avoid the Type 1 error, then we must be careful about rejecting a hypothesis that really is true. Thus we are led by the need to risk-manage the Type 1 error to widen the interval of acceptance. However, the wide acceptance criterion lets in the Type 2 error! Remember that Type 2 is accepting a hypothesis that is really false. There is no absolute rule here. It is all about experience and heuristics. Some say that the interval of acceptance should never be greater than 10%, or at most 20%. Each project team will have to decide for itself.

In many practical situations there is no bias toward optimism or pessimism. Our environmental example could be of this type, though regulatory agencies usually have a bias one way or the other. Nevertheless, if there is no bias, or it is "reasonably" small, then we know the distribution of values of H(0) or H(1) is going to be symmetrical even though we do not know the exact distribution. However, we get some help here as well. Recall the Central Limit Theorem: regardless of the actual distribution, over a very large number of trials the average outcome distribution will be Normal. Thus, the project manager can refer to the Normal distribution to estimate the confidence that goes along with an acceptance interval and thereby manage the risk of the Type 1 error. For instance, we know that only about 4%

of all outcomes lie more than ±2s from the mean value of a Normal distribution. In other words, we are about 96% confident that an outcome will be within ±2s of the mean. If the mean and variance (and from variance the standard deviation can be calculated) can be estimated from simulation, then the project manager can get a handle on the Type 1 error (rejecting something that is actually true). Figure 8-7 illustrates the points we are making.

Hypothesis H(0) NULL Hypothesis H(1) Alternate

Standard Normal Distribution Standard Normal Distribution

Standard Normal Distribution Standard Normal Distribution

ii5t?eiiigTRUE its being TRUE

Figure 8-7: Type 1 and 2 Errors.

ii5t?eiiigTRUE its being TRUE

Figure 8-7: Type 1 and 2 Errors.