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.

Decision rule: Reject H0 if \(y = 7\) or fewer fail
The significance level (probability of a Type I error) can be found from the binomial distribution of the number failing.
\[ \begin{align} P(\text{Type I error}) \;&=\; P\left(\text {decide }H_A \text{ is true} \mid H_0\right) \\[0.4em] &=\; P(X \le 7 \mid \pi = 0.06) \\ &=\; \sum_{x=0}^7 {200 \choose x} \; 0.06^x \; 0.94^{200-x} \\ &=\; 0.083 \end{align} \]
Decision rule: Reject H0 if \(y = 6\) or fewer fail
With this decision rule, the significance level is
\[ \begin{align} P(\text{Type I error}) \;&=\; P(X \le 6 \mid \pi = 0.06) \\ &=\; \sum_{x=0}^6 {200 \choose x} \; 0.06^x \; 0.94^{200-x} \\ &=\; 0.041 \end{align} \]

Since there are no intermediate decision rules, it is impossible to conduct a hypothesis test based on these data 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.

Failure of printed circuit boards

If a test with significance level \(\alpha = 0.05\) was required for the circuit board scenario, we should only decide to reject the null hypothesis (and conclude that the new production method method is better) if 6 or fewer boards fail.