Discrete distributions

For continuous distributions, we can usually determine the critical value for a decision rule that exactly corresponds to any required significance level. Unfortunately this is usually impossible when the data come from discrete distributions.

Failure of printed circuit boards

Suppose a manufacturer of a certain printed circuit has found that the probability of a board failing is \(\pi = 0.06\), and an engineer suggests some changes to the production process that might reduce this probability. Suppose that \(n = 200\) circuits will be produced using the proposed new method and \(Y\) of these will fail. We will use the sample number failing, \(y\), to test the hypotheses

We now consider two possible decision rules. Their significance levels can be found by adding probabilities from the \(\BinomDistn(n=200, \pi=0.06)\) distribution.

Decision rule: Reject H0 if \(y = 7\) or fewer fail
\[ P(\text{Type I error}) \;=\; 0.083 \]
Decision rule: Reject H0 if \(y = 6\) or fewer fail
\[ P(\text{Type I error}) \;=\; 0.041 \]

Since there are no intermediate decision rules, there is no decision rule with a significance level of exactly 0.05.

If a hypothesis test is required at a specified significance level, such as 0.05, a conservative approach should be taken. The decision rule should be chosen such that its significance level is under the required value. For example, we should reject the null hypothesis (and conclude that the new production method method is better) if 6 or fewer boards fail in the example above.